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The Theory of 

GENERAL RELATIVITY 

and 

GRAVITATION 



Based on a course of lectures delivered at the Conference on 

Recent Advances in Physics held at the University 

of Toronto, in January, 1921 



LUDWIK SILBERSTEIN, Ph.D. 




New York 

D. VAN NOSTRAND COMPANY 

Eight Warren Street 

1922 



Copyright, Canada, 1922 

BY THE 

University of Toronto Press 



io 8 no 

R 

i Musics 

S575 



PREFACE 

At the Conference on Recent Advances in Physics held 

in the Physics Laboratory of the University of Toronto from 
January 5 to 26, 1921, a course on Einstein's Relativity and 

Gravitation Theory, consisting of fifteen lectures and two 
colloquia, was delivered by the author. The first six of 1 1 ■ 
lectures were devoted to what is known as Special Relativity, 
and the remaining ones to Einstein's General Relativity and 
Gravitation Theory and to relativistic Electromagnetism. 
In view of the time limitations only the essentials of these 
theories were dealt with, due attention, however, being given 
to the critically conceptual side of the subject. The Uniwr- 
sity was kind enough to undertake the publication of that 
part of the course which dealt with general relativistic ques- 
tions, on the express understanding that my prospective 
readers should be assumed to be already familiar with the 
special theory of relativity. In this connection it was sug- 
gested by Prof. McLennan that those unacquainted with the 
older theory should be referred to my book of 1914 (The 
Theory of Relativity, Macmillan, London) and that it would 
therefore be desirable to make the present volume, as much 
as possible, uniform in exposition and style with that work. 
\Vith such requirements in view this little book was shaped, 
only a few pages at the beginning having been used in re- 
calling the essentials of the special relativity theory. 

The treatment, as compared with the Toronto lectures, 
has been made somewhat more systematic and the subject 
matter has, here and there, been considerably extended. 
In this respect the author has been partly influenced by a 
larger course on Relativity, Gravitation and Electromagnetism 
delivered, in the time of writing, during the last Summer 
Quarter at the University of Chicago. Such is especially 
the case with Chapter III in which care has been taken to 
give the readers a systematic exposition of the calculus of 
generally covariant beings called Tensors. The exposition 
follows here mainly upon Einstein's own presentation of the 
subject, with the difference, however, that due emphasis 
has been laid upon the distinction between metrical and non- 
metrical properties of tensors. But even in this chapter 



technicalities have been avoided, stress being laid upon the 
guiding principles of this new, or rather newly revived, and 
most powerful mathematical method. It seems hard to say 
whether Einstein's admirable theory has or has not a long 
life before it in the domain of Physics proper. But indepen- 
dently of its fate the time applied for studying the Tensor 
Calculus and acquiring some skill in handling it will be well 
spent. 

The plan of the remainder of the book will be sufficiently 
clear from the titles of the chapters and sections arrayed in 
the table of Contents. Such matter as seemed for the present 
too speculative and controversial has been relegated to the x 
Appendix where, however, also some points concerning the 
curvature properties of a manifold have found their place, 
not only as a preparatory to Einstein's cosmological specu- 
lations but perhaps as a useful supplement to Chapter III. 

The book is felt to be far from being complete. But as 
it is, it is hoped that it will give the reader a good insight into 
the guiding spirit of Einstein's general relativity and gravita- 
tion theory and enable him to follow without serious diffi- 
culties the deeper investigations and the more special and 
extended developments given in the large and rapidly growing 
number of papers on the subject. 

Some of my readers will miss, perhaps, in this volume 
the enthusiastic tone which usually permeates the books and 
pamphlets that have been written on the subject (with a 
notable exception of Einstein's own writings). Yet the 
author is the last man to be blind to the admirable boldness 
and the severe architectonic beauty of Einstein's theory. 
But it has seemed that beauties of such a kind are rather 
enhanced than obscured by the adoption of a sober tone 
and an apparently cold form of presentation. 

My thanks are due to Sir Robert Falconer and to Prof. 
J. C. McLennan for promoting the cause of this publication, 
to Prof. R. A. Millikan and Prof. Henry G. Gale of the 
University of Chicago for reading part of the proofs, and to 
the University of Toronto Press for the care bestowed on 
my work. 

L. S. 
Rochester, N.Y. 

November 1921. 



CONTENTS 

CHAPTER I 

Special Relativity recalled. Foundations of General Relativity and 
Gravitation Theory 

PAGE 

1. Inertia] reference systems 1 

2. Special relativity principle - 

3. Principle of constant light velocity 

4. Lorentz transformation. Galilean line-element. Minimal lines 

and geodesies representing light propagation and motion ot 

free particles 1 

5. Transition to general relativity and gravitation theory. Infini- 

tesimal equivalence hypothesis and local coordinates 9 

6. Gaussian coordinates, and the general line-element 14 

7. Light propagation and free-particle motion expressed by the 

general null-lines and geodesies 17 

CHAPTER II 

The General Relativity Principle. Minimal Lines and Geodesies. 

Examples. Newton's Equations of Motion as an Approximation 

8. Principles of general relativity; general covariance of laws. . . . 22 

9. Local and system-velocity of light 25 

10. Developed form of geodesies. Christoffel symbols 26 

10a. First example: galilean system 29 

10b. Second example: rotating system 30 

11. Geodesies, and Newtonian equations of motion as an approxi- 

mation 35 

CHAPTER III 

Elements of Tensor Algebra and Analysis 

12. Introductory. Gaussian coordinates .39 

13. Contravariant and covariant tensors of rank one or vectors . . . 40 

14. Inner or scalar product of two vectors. Zero rank tensors, 

invariants i'2 

15. Outer product. Tensors of rank two, symmetrical and anti- 

symmetrical. Mixed tensors 43 

16. Tensors of any rank 46 

17. Contraction. Intrinsic invariants 46 

18. Inner multiplication. Differentiation of tensors 48 

19. Tensor properties in a metrical field. Quadratic differential form 

or line-element 50 

20. Fundamental tensor. Metrical properties of tensors. Norm 

and size. Conjugate tensors 52 

21. Supplement. Reduced tensor 55 

22. Angle and volume. Sub-domains 56 



PAGE 

23.^Differentiation based on metrics. Covariant derivative or ex- 
pansion; contra variant derivative. Rotation of a vector. 
Antisymmetric expansion of a six-vector. Divergence of a 
six-and of a four-vector 59 

24. The Riemann-Christoffel tensor. Riemannian symbols and 

curvature. Lipschitz's theorem 62 

CHAPTER IV 
The Gravitational Field-equations, and the Tensor of Matter 

25. Contracted curvature tensor. Einstein's field-equations outside 

$$ of matter. Bianchi's identities 69 

26. Laplace's equation, and Newton's law, as a first approximation 72 

27. The tensor of matter. Einstein's field-equations within matter. 

Laplace-Poisson's equation as a first approximation. Mean 
curvature and density of matter. Example 74 

28. Equations of Matter. The principles of momentum and of 

energy. Remarks on conservation 81 

29. Hamiltonian Principle 88 

30. Gravitational waves. Einstein's approximate integration of the 

field-equations 90 

CHAPTER V 

Radially Symmetric Field. Perihelion Motion, Bending of Rays, 

and Spectrum Shift 

31. Radially symmetrical solution of the field-equations 92 

32. Perihelion of a planet. Mercury's excess 95 

33. Deflection of light rays. Results of the Sobral Eclipse Expedi- 

tion 100 

34. Shift of spectrum lines. The atoms as 'natural clocks' 102 

CHAPTER VI 

35. Generally covariant form of the equations of the electromag- 

netic field 106 

36. The four-potential Ill 

37. Orthogonal curvilinear coordinates 113 

38. Propagation of electromagnetic waves in a gravitational field . . 115 

39. Ponderomotive force and energy tensor of the electromagnetic 

field 119 

APPENDIX 

A. Manifolds of Constant Curvature 124 

B. Einstein's New Field-Equations and Elliptic Space 129 

C. Space-Time according to de Sitter 135 

D. Gravitational Fields and Electrons 137 

Index 138 



CHAPTER I. 

Special Relativity recalled. Foundations of General 
Relativity and Gravitation Theory. 



In accordance with the purpose and the origin of this 
volume* its readers are assumed to have already made them- 
selves familiar with the essentials of Einstein's older or special 
Relativity. It will be enough, therefore, to recall here very 
concisely what of that theory may be conducive to, and even 
necessary for, a thorough grasping of the structure and the 
aims of the more general theory, and of the spirit pervading it. 

1. First of all, then, out of all thinkable reference- frame- 
works, the special relativity is concerned only with a certain 
privileged class of frameworks or systems of reference, the 
inertial systems. Of these there are <» 3 . If S, say the 
'fixed-stars' system, is one of them, any other rigid system S' 
of coordinate axes moving relatively to 5 with any uniform 
and purely translational velocity v, in any direction whatever, 
is again an inertial system or belongs to the same privileged 
class. And the systems thus derived from S, or from one 
another,** exhaust the class. Since the size or absolute 
value of the relative velocity implies one scalar datum, and its 
direction two more such data, all independent of one another, 
there is just a triple infinity of inertial systems.f as already 
stated. Not that the special relativity theory abstains from 
considering accelerated, i.e. non-uniform motion of particles 
within any of these systems; but it does not contemplate any 
frameworks other than the inertial ones as systems of refer- 

*Cf. Preface. 

**If S r moves uniformly with respect to 5, and S" with respect to S', so 
does 5 with respect to 5. If the reader so desires, he may consider this as 
a postulate. 

fThe purely spatial orientation of the axes, implying further free data, 
is irrelevant in the present connection. 

1 



2 ' Relativity and Gravitation 

ence, and cannot, nor does it propose to deal with them. It 
is unable, for instance, to transform the course of phenomena 
from the 5 system to the spinning Earth or to an accelerated 
carriage as reference systems. 

2. Keeping this in mind, the first main assumption of the 
older theory, known as the Special Relativity Principle, can 
be briefly stated by saying that it requires the laws of physical 
phenomena to be the same whether they are referred to one or 
to any other inertial system. In short, the maxim of the 
1905 — Relativity was: Equal laws for all inertial systems. 

The italicized words are, mathematically speaking, at 
first somewhat vague. In fact, they are intended to stand 
for 'the same form of mathematical equations expressing the 
laws.' Now, since this implies the use of some magnitudes, 
such as the coordinates and the time, or the electric and the 
magnetic vectors (forces), in each of the said systems, the 
requirement of mathematical 'sameness' remains cloudy until 
we are told what dictionary is to be used to translate the 
language of one into that of any other inertial system, or 
technically, to transform from the non-dashed to the dashed 
variables. This vagueness, however, soon disappears, giving 
place to precision, in the next fundamental step of the theory 
as will be seen presently. 

The attentive reader might here object by saying that 'sameness of 
laws' means absence of difference, absence of observable different behaviour 
(of moving bodies or of electric waves) in passing from an S to an 5 , and 
that, therefore, to begin with, no mathematical magnitudes or equations 
are required. But actually we are, perhaps forever, confined to one 
(approximately) inertial system, our planet, and are thus unable to 
observe directly the permanence of behaviour in passing to another 
system of reference. The only way open to us is to proceed, through 
more or less long chains of abstract reasoning, from the principle of 
relativity to some observable prediction, and such processes are scarcely 
practicable without the use of mathematical symbols and equations. 

3. The second assumption, called the Principle of Constant 
Light- velocity, apart from its own importance, provides for the 
need just explained, its true office in the structure of the 
theory being to set an example of a 'physical law' which is 
postulated to satisfy rigorously the first assumption. It 



Constani Light Velocity 

nms thus: Liu' 1 ' is propagated, in vacuo, relatively 
inertial system, with a velocity c, con d equal for 

all directions, no matter whether the source emitting it is 
fixed or moving with reaped to that system. This is Bhortl) 

referred to as uniform and isotropic light propagation in an) 
inertial system. The light velocity, in empty space, plays th<- 
part of a universal constant, — which role, however, it will 
readily give up in generalized relativity. 

The reader is well acquainted with the mathematical 
expression of the consequence of these two assumptions 
(together with a tacit requirement of formal equivalence of 
any two inertial systems S, S'), to wit, the invariance of the 
quadratic form 

c 2 t 2 -x 2 -y--z 2 , 

where x, y, z are the cartesian co-ordinates and / the time of 
the ^-system. That is to say, if x', y', 2', t' be the cartesian 
co-ordinates and the time used in any other inertial system 
S', (a) should transform into 

c i r--x n --y" l -z'-. (a') 

As a matter of fact, what was originally required was that tin- 
equation (a)=0 should transform into (a') = 0, and this 
would be satisfied by putting (a') =\ . (a), where X is inde- 
pendentof x, y, z, t but might be some function of v, the relative 
velocity of S, S'. This, however, would amount to a dis- 
tinction between the two systems, at least a formal one. 
unless X=l. If, therefore, equal rights are claimed not 
only physically but also formally, mathematically, for all 
inertial systems, we have (a) = (a'), that is to say, the quadratic 
form (a) is raised to the dignity of an invariant. 

There is, certainly, nothing to object to in such a procedure, 
especially as it carries simplicity with itself. Yet these 
remarks did not seem superfluous, especially as there is among 
the relativists a strong tendency to a certain kind of hypostasy 
of the said quadatic form* (by declaring it to be more 

intensified more recently in the case of the more general (differ- 
ential) quadratic form playing a fundamental r61e in the newer relativity 
theory, as will be ?een hereafter. 



4 Relativity and Gravitation 

'objective, real or intrinsic' than space-distance or time) just 
because it "is" invariant, — and forgetting that we have 
deliberately made it invariant. 

4. Meanwhile, returning to the quadratic expression (a), 
let us write it down for a pair of events infinitesimally near 
to one another in space and time. Thus, writing X\, x 2 , x 3 , x* 
for x, y, z, ct, the statements made above can be expressed 
by saying that the quadratic differential form 

ds 2 = dxi 2 — dxi 2 —dx2 2 —dxz z (1) 

should be invariant with respect to the passage from one 
inertial system 5 to any other such system S f . The differ- 
ential foim is here preferable to the original one, as it will be 
helpful in paving the way for general relativity. 

As is well known, this requirement of invariance gives the 
rule of transformation of the variables x t into those x\ of the 
^'-system, called the Lorentz transformation. If both the Xi 
and the x\ axes are taken along the line of motion of S' 
relatively to S, with the velocity v = /3c, if further the x 2 , x s — 
axes are taken parallel to those of x' 2 , x'z, and if the convention 
x'x = x' 4=0 for xi = x 4 = is adopted, the Lorentz transforma- 
tion assumes the familiar form 

x'i = y(x! — j8x 4 ), x' 2 = x 2 , x'z = x 3 , x' i = y(x i — px 1 ) (2) 

where y — (1 — /3 2 ) ~~**. Vice versa, we have, by solving (2), 

Xi = y(x'i+t3x'^, x 2 =x' 2 , Xz = x' z , Xi = y{x' i+fix'i), 

showing the complete (including the formal) equivalence of the 
two systems. Let us keep well in mind, for what is to follow, 
that this transformation is a linear one, with constant co- 
efficients, and that special relativity, concerned with inertial 
systems only, does not contemplate any other space-time 
transformations. 

Every tetrad of magnitudes X L (t = l to 4) which are 
transformed as the x t , is called a four-vector or, after Min- 
kowski, a world-vector of the first kind. Such four-vectors 
are, in addition to dx t or x L itself, their prototype, the four- 
velocity dxjds and the four-acceleration of a moving particle, 
the electric four-current, and so on. To every vector X t 



LORENTZ TrANSFOKMTAIO F> 

belongs a scalar or invariant X\ l — X 2 — X-? — X£, its only 
invariant with respect to the Lorentz transformation. Bui 
we need not stop here to reconsider the properties of the four- 
vector and other world-vectors, such as the six-vector, which 
constituted the only lawful material of the older relativity 
for writing down laws of Nature, — especially as we shall soon 
return to these mathematical entities as particular cases of 
tensors of various ranks which are indispensable to the general 
theory of relativity. 

On the other hand we may profitably dwell yet a while 
upon the quadratic form (1) itself, the square of the line- 
element, as ds is called. Granted the assumptions of special 
relativity, this expression becomes the fundamental quadratic 
differential form of the four-manifold, the world or space - 
time, in exactly the same way as 

da 2 = dx 2 -\-dy 2 

is the fundamental form of a flat two-space or surface, and 
more generally, 

da 2 = Edu 2 >-{-2Fdudv-\-Gdv 2 

that of any surface, and 

d<r 2 =dr 2 +R 2 sin 2 ^ (sin 2 <£ d6 2 +d<j> 2 ) 

the fundamental differential form of any three-space of con- 
stant curvature i?~~.f Now, it is enough to open any book 
on differential geometry to see that, with the usual assump- 
tions of continuity, etc., the whole geometry, i.e., all metrical 
properties of the two-space or the three-space in question 
are completely determined by the corresponding differential 
forms. Their geodesies or, within restricted regions at least, 
their shortest lines, the angle relations, and their whole 
trigometry, all this is fully determined provided the co- 
efficients of the differentials, such as E, etc., appearing in 



fAccording as R 2 is positive, zero or negative, we have an elliptic, 
:lidean (or parabolic) or hyperbolic s 
becomes R sinh (r/R), where R 2 = — R 2 . 



euclidean (or parabolic) or hyperbolic space. In the latter case R sin - 

R 



6 Relativity and Gravitation 

the fundamental form are given functions of the variables.* 
This deterministic mastery of the quadratic differential 
form has been, as far back as 1860, technically extended to 
spaces or manifolds of four and, in fact, of any number of 
dimensions, — although, not being sufficiently sensational, it 
never attracted the attention of anybody beyond a few 
specialists. 

In much the same way all the metrical properties of the 
four-dimensional world of the special relativist should be, 
and are, derivable from the fundamental form (1) belonging, 
or rather allotted to it. This is, from the point of view of the 
poly-dimensional differential geometer, but a very special, in 
fact, the most simple quadratic form in four variables. For 
it contains but the squares of their differentials, and the 
coefficients of these are all constant, which — in view of the 
sequel — it may be well to bring into evidence by writing (1) 

ds 2 = g LK dx L dx K , (la) 

to be summed over t, k = 1, 2, 3, 4, tabulating the co- 
efficients, thus 

-10 

0-1 (16) 

0-1 

1 

and calling this array of special coefficients the inertial or 
the galilean g lK . We shall denote them in the sequel by g lK . 
To give this array is as much as to give the form (la), and 
herewith the properties of the world, — for it is manifestly 
irrelevant how we call or denote the four corresponding 
variables. The values of the g lK being given, the properties 

*To be rigorous we should have said ' all properties of a restricted region 
of the contemplated manifold'; for certain properties of the manifold as 
a whole are still left free. The choice, however, is limited to a small number 
of discrete possibilities. Thus, for example, there are two kinds of elliptic 
space, the spherical or antipodal, and the polar or elliptic proper. In the 
former the total length of a straight line (geodesic) is 2tR, and in the latter 
irR; the planes are two-sided, and one-sided, respectively, and so on. 



Minimal Links AND GeODI « 

of the x will follow by themselves. There i no need to 
declare beforehand thai they are cartesian coordinates of a 
place and its date. Further, the circumstance that th< 

coefficients arc of different signs, three being negative, and 
one positive, creates for the general geometer no difficulty. 

This circumstance brings only with it the important 
feature that there are in the world real minimal lines* as the 
geometer would put it, that is to say, lines of zero-lenytli. 



ds = o. 



or 

dxi 2 + dx2*-\- dx^ 
dxi 1 



c 2 \dis 



1. 



These special world lines represent the propagation of light 
or, apart from physical difficulties, the uniform motion of a 
particle with light velocity c. As a matter of fact the very 
first step of the theory consisted in writing ds = as the ex- 
pression of light propagation in vacuo. 

In the next place consider the equally fundamental con- 
cept of the geodesies of the world. These are defined by 

8/ds = o. 

the limits of the integral being kept fixed. To derive from 
this variational equation the differential equations of a 
geodesic, proceed in the well-known way. If u be any inde- 
pendent parameter, and if dots are used for the derivatives 
with respect to it, we have 

8fsdu=J8s . du — o. 

where, by (1), s~= — (*i 2 +.^ 2 +*3 2 )+*4 2 > and therefore, 

8 s = -r (x 4 5^4 — £i5xi— x 2 5.v" 2 — £35x3). 
5 



*Whereas on any (real) surface all the 'minimal lines' (known also as 
null-lines), which play in the surface theory an important analytical r61e, 
are always imaginary. The reader will do well to consult on this and allied 
topics a special treatise on differential geometry. 



8 Relativity and Gravitation 

By partial integration, and remembering that all 8x t vanish 
at the limits of the integral, 



- xfix,, . du— — \ — ( — x t )8x t . du. 
s J du\ s / 



Thus, the 8x t being mutually independent, the required 
differential equations are 



©=' 



d : : .0. 

du 



If the geodesic does not happen to be a null-line (light pro- 
pagation) we can as well take u = s, when s=l, and the 
equations become 

ds 2 
whence 

Q/OC. (jsdC, Q/jO^ 

— = = a t = const. 

ds dxi ds 

The fourth of these equations is dxt/ds = const., and therefore, 
the first three, 

dxi _ dx 2 _ dx 3 _ 

— ^i> — °2> — o-z, 

dt dt dt 

and these represent uniform rectilinear motion, which is the 
motion of a free particle. 

Let us, therefore, keep well in mind these two properties 
of the line-element ds of special relativity: 

I. The minimal lines of the world, 

ds = 0, (I) 

represent light propagation in vacuo. 

II. The world geodesies, defined by 

5fds = 0, (II) 

with fixed integral limts, represent the motion of a free particle. 



Minimal Lines and Gbodb 9 

A special emphasis is here put on these two prop 
because they will be carried over to the general relativity 

and gravitation theory, and because these and principally 
only these two properties constitute the connection of the 
otherwise purely analytical differential form ds 2 = g^dx^dx, 
with physics. In other words, (I) as the equation of light 
propagation, and (II) as that of the motion of a free particle 
impart physical meaning to the mathematical form which 
is the 'line-element' ds. Without this all the properties of 
the quadratic form, though interesting, perhaps, in them- 
selves, would have nothing to do with the world of physical 
phenomena. 

It is scarcely necessary to say that the law (II) of the 
motion of free particles is, as well as (I) for light, invariant 
(thus far) with respect to the Lorentz transformation. For 
it is, by its very structure, independent of the choice of a 
reference system 5. Since ds is invariant, so \sfds, extended 
between any two world-points. Thus also the developed 
form of (II), the system of differential equations d 2 xjds- = 0, 
is transformed in S' into dhc l '/ds 2 = 0. And in fact, uniform 
motion of a particle relatively to S, means also (originally by 
an assumption) its uniform motion in any other inertial 
system S'. In short, the Lorentz transformation leaves the 
uniformity of motion of a particle intact. 

5. We are now ready to pass to Einstein's theory of 
general relativity and gravitation. Not that our task is an 
easy one, but we are somewhat better prepared to embark 
upon it. 

Why equal form of physical laws, why equal rights for 
the inertial systems only? Why not equal rights for all 
(systems)? Such would be the urgent, and yet vague, ques- 
tions naturally suggesting themselves after what was said 
in the preceding sections. Yet it is not with these questions, 
nor with an attempt to answer them, that we will begin our 
journey across this new and revolutionary country. For, 
even if answered, these questions would remain physically 
barren were it not for the existence of gravitation and 



10 Relativity and Gravitation 

especially of a certain peculiarly simple property of this 
universal agent. 

This, therefore, will first occupy our attention for a while. 
The cardinal feature of gravitation just hinted at is the pro- 
portionality of weight to mass, in other words, the proportion- 
ality of heavy (gravitating) and inert mass. First tested by 
Newton in his famous pendulum experiments with bobs of 
different material, and carried to further precision by Bessel, 
this proportionality has been more recently shown by Roland 
Eotvos to hold to one part in ten millions. It is reasonable, 
therefore, to assume, with Einstein, that it holds rigorously,* 
at least until proofs to the contrary are forthcoming. In our 
present connection it is better to express this property more 
directly by saying, even with Galileo, that all bodies, light 
or heavy, fall equally in vacuo. All particles, that is, acquire 
at a given place of a gravitational field equal accelerations 
independently of their own mass or chemical nature, etc., and 
no matter how much of their inertia is due to the energy 
stored in them and how much of other origin. This remark- 
able property distinguishes the gravitational field from other 
fields. Take, for instance, an electric field given by the vector 
E. The force on a particle of rest-mass m, carrying the 
electric charge e, and starting from rest, is eE, and the accelera- 
tion eE/m. Now, in general, there is no relation between m 
and e, and even if the mass is purely electromagnetic, when m 
is proportional to e 2 /a, the acceleration will vary from particle 
to particle inversely as its charge and directly as its average 
diameter, 2a. We have disregarded, of course, the dielectric 
properties of the particle which would make its behaviour 
in a given electric field still more complicated. The same 
remarks would hold, mutatis mutandis, for the behaviour of 
different bodies placed in a magnetic field. In short, gravita- 
tion is, in this respect, unique in its simplicity. 



*In a theory of matter and gravitation proposed by G. Mie, Annalen 
der Physik, vols. 37, 39, 40 (1912 and 1913), the proportionality between 
weight and mass does not hold rigorously, though to an order of precision 
much exceeding that stated by Eotvos. 



Equivalence Hypothj 11 

This very circumstance enabled Einstein to undertake his 
mental experiment with the falling or ascending elevator, 

now so familiar to the general public. In fart, consider a 
homogeneous or a quasi-homogeneous gravitational field 
such as the terrestrial one in a properly restricted region. Lei 
a lift or elevator, small compared with the earth, yet ample 
enough for a physical laboratory and for those in charge of 
it, descend vertically with the local terrestrial acceleration g. 
Then all bodies placed anywhere within the elevator and 
left to themselves will float, in mid-air or better in vacuo, 
and particles projected in any direction will move uniformly 
in straight paths relatively to the elevator. Moreover, all 
objects, including the physicists, standing or lying about 
will cease to press against the floor or the tables, as the case 
may be. In short, all traces of gravitation will be gone,* 
and the inmates of the lift, assumed to have no intercourse 
whatever with the outer world, will declare their reference 
system to be a genuine inertial system, — so far, at least, as 
mechanical phenomena are concerned. For an unbiassed 
judge could not tell beforehand whether it will be also optically 
inertial, that is to say, whether the law of constant light 
velocity will hold good for the lift. Einstein thinks it will, or 
rather assumes it, more or less implicitly. If this be granted, 
we can say that the elevator will be an inertial reference 
system in every respect. 

The possibility of thus undoing a gravitational field is 
manifestly based on the said equal behaviour of all bodies 
placed in it. For otherwise the artificial motion of the elevator 
could not be adapted to all bodies at the same time, each of 
them requiring a different acceleration. 

Next, pass to any, non-homogeneous gravitation field, 
which in the most general case may also vary with time. This 
certainly cannot be undone, as a whole, by a single elevator 
as reference system. But you can imagine an ever increasing 
number of sufficiently small elevators, each appropriately 
accelerated, fitted into small regions of the field, and each, 

*Vice versa, in absence of a gravitational field, a lift in accelerated 
ascending motion would give us a faithful imitation of such a field. 

—2 



12 Relativity and Gravitation 

perhaps, to do its duty for a very short interval of time, and 
to be replaced by another in the next moment. These minute 
elevators will do their office at least in the mechanical sense 
of the word. Einstein assumes that they will act as inertial 
systems also in the optical sense of the word, as explained 
above. This process of subdividing a gravitational field, in 
space and time, and fitting in of appropriate small elevators 
can be carried on to any required degree of approximation. 

In fine, passing to the limit, let us make, with Einstein,* 
the explicit assumption: 

With an appropriate choice of a local reference system 
(ui, u 2 , u z , Ui) special relativity holds for every infinitesimal 
four-dimensional domain or volume-element of the world. 

That is to say, at every world-pointf a system of space- 
time coordinates u x , u 2 , u s , u± can be chosen in which the line- 
element assumes the galilean form 

ds 2 = du£ — du\—du 2 2 — du£. (3) 

These four coordinates are called local coordinates. With 
respect to this local system there is then no gravitational field 
at the given world-point, and in accordance with special 
relativity ds 2 has there a value independent of the 'orienta- 
tion ' of the local axes; that is to say, the quadratic form (3) 
is invariant with respect to the Lorentz transformation (2). 

It is this assumption which can now be properly referred 
to as the infinitesimal equivalence hypothesis, for it grew out 
of Einstein's original equivalence hypothesis applied to finite 
regions when, in his first attempt at a theory of gravitation 
(1911), he was confining himself to a homogeneous field. 

Whatever the origin of this hypothesis or assumption, it is 
certainly not difficult to adhere to it. For it scarcely amounts 
to anything more than to assuming, in the case of a curved 
surface, say, the existence of a tangential plane at any of its 



*A. Einstein, Die Grundlagen der allgemeinen Relativitatstheorie . 
Annalen der Physik, vol. 49, 1916, p. 777. 

fWith the possible exception of some discrete points, such perhaps 
as those at which the density of matter acquires enormous values. 



Infinitesimal World Flatni i 3 

points, or to <lc( lare the surface to be (in Clifford*! termino- 
logy) elementally flat. And it will, perhaps, I"- wt 11 to 
shortly Einstein's hypothesis by Baying thai it the 

jour-dimensional "world to he, in presence as well a- in ab • 
of gravitation, elementally /Jul. It will not be forgotten, I 
over, that this geometric term is nothing more than a synonym 
of elementally galilean, i.e., satisfying special relativity in- 
finitesimally.* 

To avoid the danger of any misconception let u- dwell 
upon this subject yet for a while. The coordinates u t with 
their corresponding galilean line-element (3) were set up 
only for a local purpose, their real office being confined to a 
fixed world-point P, say x u x 2 , x 3 , x* (in any coordinate 
system). If we so desire, we may think of a whole galilean 
world U determined throughout, to any extent, by the simple 
form (3). But as a tangential plane has something in common 
with a surface only at the point of contact and then diverges 
from it, ceasing to represent any intrinsic properties of the 
surface itself, so has the auxiliary and fictitious world I' 
anything to do with the actual world W (complicated by 
gravitation) at the world point x t only. The fictitious world 
Uis tangential to the actual world IF at that point, and parts 
company with it beyond the point of contact. At other 
world-points the role of U is taken over by other and other 
fictitious galilean worlds. One more cautious remark. The 
contact of U and W is one of the first order, i.e., such as 
the contact between a surface and its tangential plane 
or between a curve and its tangential line, but not as the 
more intimate contact bewteen a curve and its circle of 
curvature (which is of the second order). This circumstance 
may acquire some importance later on. 



*As to the concept of elementary flatness of a surface or a more-dimen- 
sional space, it is beautifully explained in W. K. Clifford's ' Philosophy of 
Pure Sciences', published in his famous Lectures and Essays (Macmillan, 
London). Notice that in Clifford's sense every regular surface, no matter 
how curved, is elementally flat, with the exception of some singular points, 
such as the vertex of a cone. 



14 Relativity and Gravitation 

6. Having thus made clear the local character of the w t 
coordinates, let us now introduce any coordinate system x 4 
whatever, to be used as a reference system of coordinates 
for the whole world, i.e., throughout the gravitational field 
and through all times. Then, if x t be the reference coordinates 
of P, and x t -\- dXt those of a neighbour-point u t -\-du lt the 
differentials du l will in general be linear homogeneous func- 
tions of all the dx L , say 

4 

(LU L —- 2j CL lK (tX K , 

or with the conventional abbreviation, 

du t = a lK dx K , (4) 

where the coefficients a lK will in general be functions of all 
the x t . It is of importance to note that that the relations (4) 
will, generally speaking, be not-integrable, or borrowing a 
name from dynamics, non-holonomous, that is to say, the 
a lK will not necessarily be duJdx K , the differential expressions 
on the right of (4) will not be total differentials of functions 
of the x t , and there will be no finite relations between the 
local and the general or reference-coordinates. 

Substituting (4) into (3), collecting the terms and calling 
g lK the coefficient of the product of differentials dx t dx K we 
shall have, for the line-element in the general reference 
system, 

ds 2 = g tK dx t dx K , (5) 

where g lK = g Kl will, in the most general case, be functions of 
all the x t . But since ds 2 , as defined originally by (3), was in- 
dependent of the orientation of the local system of axes, so 
also will the ten different coefficients g lK , though functions of 
the coordinates x t , be manifestly independent of the orienta- 
tion of the local system. 

The line-element will thus be represented, in any reference 
coordinates x t whatever by the most general quadratic differen- 
tial form of these four variables, such in fact being the form 
(5). As before the summation sign is omitted; the sum- 
mation is to be extended over i, k from 1 to 4, each of these 



Cknkkal Tkansfomiati 



15 



suffixes, l and k, appearing twice. Thus, ds 2 = i[ndx l *+ 
2g u dx\dxz-\- . . . +f>udxi 2 . The reader will soon learn to 
handle this abbreviated and very convenient symbolism. 

Suppose now we introrluce instead of X t any other 
space-time coordinates x/, any functions whatever "I il i 
such, that is, that between the two sets exist any given holo- 
nomous relations 

*i=&(*i', x 2 ', x 3 \ xS), (6) 

the <f> t being any functions whatever, but continuous together 
with their first derivatives and such that their Jacobian, the 
well-known determinant 



J 



dx t 



dxj 



dx\ dxi 
dx\ dx 2 ' 


dx\ 
dXi 




dx 4 dXi 


dXi 



dx\ 



(7) 



dxi' dxo' 
does not vanish. Under these circumstances we have 

dx t = — -dx K ' ', 
dxj 



and vice versa, 



dxj 
dx/ = — - dx K , 
dx r 



(8) 
(8a) 



and, as it may be well to notice in passing, //'= 1, where /' 

dx{ 



is the inverse Jacobian 



dx M 



Now, substituting (8) into 



(5), gathering again the terms, and denoting the coefficient of 
dx t 'dx K ' by gj , i.e., putting 



, dXa chj, 
dx/ d.v, 



- &afi, 



we shall have 



ds 2 = g LK f dx l 'dx K ' 



(9) 



(50 



which is (5) reproduced in dashed letters. Not that the gj 
will be functions of the x/ of the same form as were the g^ of 



16 Relativity and Gravitation 

x„ but only that the quadratic differential form remains 
quadratic. There is certainly nothing surprising in this kind 
of permanence.* Yet this and this only is justifiably meant 
when we say that the line-element ds 2 is invariant with 
respect to any transformations whatever. If the relativist 
sees anything more in "the invariance of ds", namely that 
ds is something belonging to a pair of world-points (x, and 
x t -\-dx t ) inherent in that pair independently of the choice of 
a reference system, it is what he puts into it at a later stage 
by ascribing to it certain physical properties, or by inter- 
preting it physically in certain ways. The meaning of these 
remarks will gradually become more intelligible. 

Before passing on to the two cardinal virtues conferred 
upon the line-element, one more mathematical remark about 
it may not be out of place just here. Suppose the line- 
element (5) is actually given with some determined and more 
or less complicated functions as the g lK . By trying, in succes- 
sion, other and other new variables xj we would arrive at a 
great variety of new forms of functions gj . The natural 
question arises: Are there not among all these sets of co- 
ordinates just such as would convert (5), throughout the 
world or a finite world-domain, into a galilean line-element, 
i.e., one with constant coefficients? The answer is, in general, 
in the negative. A given form ds 2 = g lK dx t dx K is equivalent, that 
is to say, can be reduced by holonomous transformations, to 
a form with constant coefficients and thus also to the galilean 
line-elementf when and only when certain differential 
expressions formed of the g^, their first and second deriva- 
tives, all vanish.* These expressions, of which more will be 

*Notice that the case is different in special relativity, where we require 
the form to reappear with all its original coefficients, three — 1, and one+1. 

fThe circumstance that three of the coefficients of this form are negative 
and one positive imposes on the original g dx dx to be thus transformable 
certain further conditions in connection with the so-called 'law of (alge- 
braic) inertia', due to Sylvester. 

*The restriction to ' holonomous transformations ' is of prime importance. 
For by means of non-holonomous or non-integrable relations, such as (3), 
every g dx t dx can be transformed into a quadratic differential form with 
constant coefficients. 



Rjemann's Symbol it 

said in the sequel, arc known in general differential 
geometry as Kiemann's four-index symbol*. Of ' 
symbols there arc in the ease of any number n of din 

— n 2 (n 2 — 1) linearly Independent ones. Thus an ordinary, 

two-dimensional, surface has hut one Riemann symbol and 
this is its Gaussian curvature, multiplied by gng22 — ga* t tin- 
determinant of the g, K . Any three-dimensional manifold has 
six, and our, or rather Einstein-Minkowski's world has as 
many as twenty linearly independent Riemann symbols. 
Thus any finite domain of the world is equivalent to a galilcan 
domain when and only when all these twenty symbols vanish 
in that domain, i.e., when the ten different £« satisfy within it 
a system of twenty partial differential equations of the second 
order. (It will be useful to keep in mind the last italics.) 
By what has just been said it is manifest that if all the Rie- 
mann symbols vanish in one system of coordinates x t , they 
will vanish also in any other %/ obtained from the former by 
any holonomous transformations whatever. 

But enough has for the present been said on the symbols 
of that great geometer. Later on they will be seen to play 
an all-important role in Einstein's gravitation theory. 

It is now time to return to the physical aspect of our 
subject. 

7. Having assumed, after Einstein, that special relativity 
holds for every infinitesimal domain, or that the world is 
elementally galilean, we wrote down the simple form (3) in 
local coordinates u t . Then, passing to any coordinates x t by 
means of the non-holonomous relations (4) w r e obtained for 
the line-element of the world the general quadratic differential 
form (5), with variable coefficients g lK , functions of the x g . 

But what is the physical meaning of this general ds with 
all its ten different g lK ? What are they to represent physically ? 
The answer is that we are still to a certain extent the masters 
of the situation, and can make them have that physical 
meaning which we will put into them. For thus far we know 
only the physical meaning of the galilean element belonging 
to a world U, and that (in virtue of an assumption^ the world 



18 Relativity and Gravitation 

W as a seat of or deformed by gravitation is galilean in its 
elements, or that at each of its points a £/"-world tangential to 
it can be constructed. 

At this stage then we are entitled only to say that (since 
^without gravitation is If, and since to If belong the constant 
coefficients g tK ) the essential differences* in the coefficients of 
the two worlds, g lK and ~g lK , are due to, or better, are somehow 
connected with gravitation. But exactly how, we cannot, 
thus far, say. For our position is somewhat like this : Suppose 
we know that a surface a, which is not a plane as a whole, is 
elementally flat and thus has a tangential plane tt at each of 
its points. Suppose further we know the physical properties 
of certain lines (straights, or circles, etc.) drawn on any ic. 
Does this alone enable us to say what the physical properties 
of similarly defined lines will be when drawn on o-? Clearly 
not. For the 7r-lines have but a single point of contact with 
a, and that only of the first order, and deviate from the 
surface or become extra-a beings all around the point of 
contact. 

Now, in the case of space-time, we fixed the physical 
meaning of the line-element of the LV-world by declaring its 
minimal lines, ds = 0, to be the law of light propagation, and 
its geodesies, 8fds = 0, to represent the motion of free particles. 
Does this, and the existence of a tangential If at every point 
of the actual world W, entitle us to assert that the minimal 
lines and the geodesies of W will again represent the optical 
and the mechanical laws in this world? This is by no means 
a superfluous question. For the auxiliary tangential world 
U leaves the actual world beyond the point of contact and 
becomes at once fictitious or extra-mundane, so to speak. 

Now, the minimal lines of t/,t defined by a differential 
equation of the first order, are also, at P, minimal lines of W, 
so that at least the starting elements of these lines are identical. 
At the next element the r61e of If is taken over by another 

*i.e. t those, at least, which cannot be abolished by holonomous co- 
ordinate transformations. 

fWhich fill out only a conic hypersurface (of three dimensions) with the 
contact point P as apex. 



GEODESICS AM) Law 01 Mom- l'» 

galilean world; yei the reasoning can be repeated, jo thai 
can say that every elemeiii of ;i minimal line "I W represents 

light propagation, and thence deduce that such a ll'-line 
possesses also as a whole the same physical property. Fait 
the position is altogether different with the geodesil I or 
these world-lines are defined by differential equations of the 
second order* so that the mere contact of U and W (bein 
the first order) does not at all entitle us to transfer any 
properties of the geodesies of U upon those of W, not even 
at their very starting point P. 

If, however, the said physical property of the K'-geodesics 
does not follow logically from the previous assumptions, yet 
we are free to introduce it as a further explicit assumption. 
In fact, while thus generalizing the physical significance of the 
geodesies Einstein is well aware that this is a new assumption,! 
although one that easily suggests itself. Nor is there any 
inconsistency in thus transfering a property from the galilean 
to the more general world-geodesies. For, as we shall see 
later on, the developed form of the equations of the geodesies 
contains only the g lK and their first derivatives with respect 
to the x„ whereas the conditions characterizing a world as 
galilean (the vanishing of the Riemann symbols) are equations 
between the g lK , their first and second derivatives, and there 
are no relations at all between the g lK and their first derivatives 
alone. 

But even with this new assumption, the total number of 
assumptions of Einstein's theory is remarkably small. And 
as to the advisability of making the one just discussed, we 
may say that Einstein's theory owes to it the greater part of 
its power. 

The property of the geodesies being thus assumed, and 
that belonging to the minimal lines being deducible from 
what preceded, we are now in the position to sum up definitely 
and very concisely, if not the whole, yet the most fundamental 
part of Einstein's theory. For this purpose we have only to 

*A geodesic issues from P in every direction whatever in the four- 
manifolds U and W. 

fA. Einstein, loc. ciL, p. 802. 



20 Relativity and Gravitation 

repeat the previous statements I. and II. without their 
restrictions, replacing the galilean ds by the general one and 
adding a few explanatory words. Thus: 

The world-line element, in any system of coordinates, and 
whether gravitation be absent or present, is given by 

ds 2 = g lK dx t dx K , (10) 

where g tK = g Kt are some functions of, in general, all the jour 
coordinates, but of these alone. If these ten functions be given, 
all metrical properties* of the world are determined, and among 
these its minimal lines, 

ds = 0, (I) 

and its geodesies, 

8fds = 0. (II) 

The physical significance of these world-lines is that the former 
represent propagation of light in vacuo, and the latter the motion 
of a free particle. 

By a 'free' particle is meant one which, having received 
any initial impulse is left to its own fate, whether in absence 
or in proximity of other lumps of matter (absence or presence 
of 'gravitation'), but not colliding with them, and in absence 
of, or better not immersed in, an electromagnetic field. One 
strives in vain to enumerate all the attributes of a concept 
which can become clear only a posteriori, through the concrete 
applications of the theory. Suffice it to say that ' free particle ' 
may as well stand for a projectile, in vacuo, or a planet 
circling around the sun. Their laws of motion are given by 
the corresponding world-geodesies. The developed form of 
the equations of the geodesies, as well as of light propagation, 
will be given later on. 

Since the g lK are to determine, through (II), the fall of 
projectiles and the motion of celestial bodies, it is scarcely 
necessary to repeat that they are intimately connected with 
gravitation. These ten coefficients will replace the unique 
scalar potential of newtonian mechanics. They will influence 

* Apart from some properties of the world as a whole, — of which more 
later on. 



Free Particles AND LlGHl J I 

also, through (I), the course of light in interplanetary ar.fi 
interstellar spaces, and finally, by t heir very appearance in 
the line-element, they will mould the geo- and ehrono-metrical 
properties of our world. These latter properties thus appear 
intimately entangled with gravitation and optics. 

It remains to explain how these all-powerful coefficients 
arc, in their turn, determined in terms of other things such 
as the density of 'matter'. This is the office of Einstein's 
'field-equations' which will occupy our attention in the 
sequel. 



CHAPTER II. 

The General Relativity Principle. Minimal Lines and 

Geodesies. Examples. Newton's Equations of 

Motion as an Approximation. 



8. Most readers will perhaps be surprised to find in the 
first chapter almost no mention of the general principle of 
relativity which claims equal rights for all systems of co- 
ordinates, and which in all publications on our subject is 
given the most prominent place. Instead of this we insisted 
on the general form of the line-element (10), on the null-lines 
and the geodesies of the world metrically determined by that 
line-element, and still more upon the physical meaning of 
these two kinds of world-lines as representing light propa- 
gation and the motion of free particles. 

The reason for adopting this plan is that, as far as I can 
see, these things are most important from the physical point 
of view, nay, they are perhaps* the only relevant constituents 
of the new theory looked upon as a physical theory. This is 
particularly true of the optical and mechanical meaning 
attributed to the said two kinds of lines, thus giving what the 
logicians call a concrete representation of what otherwise 
would be only a purely mathematical or logical science, an 
abstract geometry of a manifold of four dimensions deter- 
mined by that quadratic differential form. It is exactly this 
physical interpretation which invests the theory with the 
power of making statements of a phenomenal content, of 
predicting the course of observable events. On the other 
hand, the much extolled Principle of General Relativity 
which, in Einstein's wording,! requires 
The general laws of Nature to be expressed by equations valid 



*Apart from ' the field equations ', yet to come. 
\Loc. cit., p. 776. 

22 



General RELATIvm PRINCIPLE 23 

in all coordinate systems, i.e., covariant with respect to any 
substitutions whatever (generally covariant), 
is by itself powerless either to predi< I or to exi lude anything 
which has a phenomenal content. For whatever we already 

know or will learn to know ahout the ways of Nature, pro- 
vided always it has some phenomenal contents (and is not a 
merely formal proposition), should always be expressible in 
a manner independent of the auxiliaries used for its descrip- 
tion. In other words, the mere requirement of general 
covariance does not exclude any phenomena or any laws of 
Nature, but only certain ways of expressing them. It does 
not at all prescribe the course of Nature but the form of the 
laws constructed by the naturalist (mathematical physicist 
or astronomer) who is about to describe it. The fact that 
some phenomenal qualities are technically (with our inherited 
mathematical apparatus) much more difficult to put into a 
generally covariant form than some others does not in the 
least change the position. 

To make my meaning plain, let us take the case of plane- 
tary motion. For the sake of simplicity let there be but a 
single planet revolving around the sun. It is well-known that 
according to Newton the orbit of the planet should be a 
conic section, say an ellipse with fixed perihelion.* It is, in 
our days, almost equally well known that according to 
Einstein's theory the perihelion should move, progressively, 
showing a shift at the completion of each of its periods. And 
so it does, at least to judge from Mercury's behaviour. At 
the same time Einstein's equations are generally covariant, 
while Newton's 'law' or Laplace-Poisson's equations are not.f 
What of this? Does it mean that fixed perihelia are excluded 
or prohibited by the principle of general covariance? Cer- 
tainly not. Provided that 'fixed perihelion' and 'moving 
perihelion' have, each, a phenomenal content, and this they 
do, both kinds of planetary behaviour should be expressible 
in a generally covariant form. Newton's inverse square law 
and his equations of motion certainly do not express it so, 

*Fixed, that is, relatively to the stars. 

fNot even with respect to the special or the Lorentz transformation. 



24 Relativity and Gravitation 

and it may be difficult to find a covariant expression for a 
strictly keplerian behaviour. But if it were urgently needed, 
some powerful mathematician would, no doubt, succeed in 
constructing it. If, as actually is the case, Einstein's theory 
excludes a fixed perihelion, and other newtonian features, it 
does this not in virtue of the said principle alone (nor even in 
part), but pre-eminently owing to the physical meaning 
ascribed to the world-geodesies, and to the choice of his field 
equations which again are physically relevant since they 
determine the g lK influencing essentially the form of those 
world-lines. That the principle of general relativity turned 
out to be helpful in guessing new laws (by limiting the choice 
of formulae) is an altogether different matter. It may prove 
an even more successful guide in the future. f But here its 
role ends, — always taking the Principle only as a mathematical 
requirement of general covariance of equations. And so it 
is at any rate enunciated (and interpreted, cf, p. 776, loc. cit.) 
by Einstein himself, although some of his exponents put into 
it a physical meaning. In fact, as we shall see later on, the 
sameness of form of the equations (of motion, say) in two 
reference systems, as in a smoothly rolling and a vehemently 
jerked car, does not at all mean sameness of phenomenal 
behaviour for the passengers of these two vehicles. 

So much in explanation of the absence of the general 
principle of relativity in all our preceding deductions. 

It will be noticed, however, that although no explicit 
mention of this principle has been made in Chapter I, yet the 
fundamental laws (I) and (II) there given do satisfy this 
principle. In fact, both the null-lines and the geodesies of 
the world were defined without the aid of any reference 
system. And as to the line-element itself, its invariance was 
seen to be automatic. 

Thus, in what precedes we have, without insisting upon 
it, been faithful to the formal principle of general relativity. 
Nor is it our intention to depart from it in what will follow. 

fOr it may become sterile to-morrow, as is the fate of almost all our 
Principles. 



Light Veixx n v 

As was already mentioned at the close o( the first chapter, 

to make the expo iiion of the fundamental pan of Einstein's 

iheory complete, il remains to add to MO;, ( I J, (I I ), together 
with their optical and mechanical meaning, a -' I of equal ion - 
determining the ten coefficients #„ of the quadratic form. 
But before passing to these differential (filiation-, Ein t< in'- 
field-equations, it will be well to discuss somewhat more and 
to develop those already given. Some explanations and 
examples concerning the transformation of coordinates will 
also be helpful at this stage. 

9. First, concerning the law of propagation of light 
(in vacuo), to obtain its developed form it is enough to sub- 
stitute the line-element (10) into the equation (I) of the 
minimal lines. Thus the fundamental optical law will be 

g lK dx l dx K = 0. (11) 

It gives the velocity of light for every direction of the ray, i.e., of 
the infinitesimal space-vector dx\, dx 2 , dx 3 , if dx\jc be the 
time element of the reference system. In general the light 
velocity will differ from c and have different values at different 
world points and for different directions of the ray. 

This "light velocity" which has nothing intrinsic about 
it is to be distinguished from the local velocity of light (that 
corresponding to a local, galilean system of coordinates) 
which is the same for all directions. To avoid confusion the 
former may be called the system-velocity of light or, according 
to some authors, the 'coordinate velocity' of light. It is a 
kind of velocity estimated from a distant standpoint. If we 
write it, in a given reference system, 

dcr _ da 
dt dXi ' 

the very concept of such a light velocity, whose value is to 
be derived from (11), presupposes that 'the length' da of the 
infinitesimal space-vector dx u dxi, dx* has been defined in 
some way for that system in terms of these differentials and 
the coefficients g lK . We shall have the best opportunity of 



26 Relativity and Gravitation 

explaining how this is done technically in deducing physical 
results, when we come to speak of the bending of rays of light 
around a massive body such as the sun. Then also the 
question will be mentioned under what circumstances the law 
of Fermat, giving the shape of the rays, is applicable. 

In the meantime it is advisable to look upon (11) as the 
equation of the infinitesimal wave surface at the instant t-\-dt 
corresponding to a light disturbance started at xi, x 2 , x s at the 
instant /, the differential dx± being treated as a constant 
parameter. From the local standpoint this surface is, of 
course, a sphere, but from the distant (or system-) standpoint 
it may have a variety of more complicated shapes. It would, 
perhaps, be rash to say that it will be a quadric. But, being 
locally closed, it may also be expected to be a closed surface 
from the system-point of view. 

10. Next for the geodesies of the world. The developed 
form of their differential equations is easily derived from their 
original definition (II), 

8fds = 0. 

As in the case of a galilean world, let u be any parameter, and 
let dots stand for derivatives with respect to it. Then 

f8s . du = 0, 
where, in the most general case, 

S 2 = gucXtX K . (12) 

The variation of s can be written 

ds « i <?$ „. 

os = 5x t + — dX t , 

dx c dx t 

to be summed over t= 1 to 4. Thus, by partial integration of 
the second terms, the limits of the integral being fixed, 



d / ds \ _ ds_ - 0, 
du V dx. s dx. 



du \ dx L / dx t 
and by (12), with 5 itself taken for u, 



Geodes h 



rl:\ tK ds / " dx. ds (Is 



or 



ds 2 dx x ds <ls dx t ds ds 

Introducing the expressions, known as Christofft l's syml 

r«/n m l /a^ a d^ii _ ***) = p G l 

LtJ V^ dx a dxy/ LtJ 

we can condense the last set of equations into 

a ^5 -u |~ a,3 ~l ~" — = n 
gut ds^L^Jds ds ' 

These are four linear equations for the four d-x K , ds"-. Let us 
solve them for these derivatives. Denoting the second term 
by a„ and writing g for the determinant of the g„, we shall have 



d 2 Xi 1 
ds 2 g 



fll g]2 gl3 gU 



a 4 g42 gi3 gii 



= 0, etc. 



or, if g uc = g'" be the minor of g, corresponding to g^, divided by 

g itself, 

— -r + «ig u + Ofig 12 + a 3 g 13 + a^ 14 = 0, etc., 
ds- 



t.e., 



</s ? * L « J ds ds 



Here we will write, after Christoffel, 



-3 



fa/3) o „ra/T| .. )o'a\ 



(14) 



28 Relativity and Gravitation 

Thus, ultimately, the differential equations of the geodesies 
or the equations of motion of a free particle will be, in any system 
of coordinates, 

(^ + {"P\dx« ^% =0 * (1 5) 

ds 2 \ >■ ) ds ds 

These are four equations. But since we have, identically, 

ds as 

one of these equations of motion is a consequence of the 
remaining three, a feature already familiar to the reader from 
special relativistic mechanics. Since these differential equa- 
tions are only the developed form of 8fds = 0, they will mani- 
festly be generally covariant, that is to say, in any new 
coordinates xj the equations (15) will be 

d 2 xj jap\'dx a ' dx/ _ 



ds 2 \ L ) ds ds 

If the coefficients g lK are all constant, all the Christoffel 

symbols < > vanish and the equations (15) reduce to 

d 2 xjds ? = 0, which represent uniform rectilinear motion. 
And since the general equations (15) represent the motion of 
a free particle in any gravitational field and in any system, the 

symbols < V, built up of the g lK and their first derivatives, 

can be said to express the deviation of the motion from 
uniformity due to gravitation, and partly due to the peculiari- 
ties of the system of reference. In view of this property, and 
disregarding any distinction between gravitation proper and 
the effects of the choice of the coordinate system, f Einstein 



*This form of the equations of a geodesic of a manifold, of any number 
of dimensions, has been used by geometers for a long time. See, for 
instance, L. Bianchi's Geometria differentiate, vol. I, Pisa 1902, p. 334. 

fOr between permanent acceleration fields and such that can be trans- 
formed away. 



Christoffel Symboi 

proposes to call these ( Ihristoffel symbols 'the components of 
the gravitational field '. 

Notice, however, that if all \ ( vanish in om 

1 ' ) 

of reference they do not necessarily vanish in other systi 

(even if obtained from the former by holonomous transforma- 
tions). In view of this circumstance the name proposed by 
Einstein seems utterly inappropriate and misleading, even 
if one agreed not to distinguish between permanent fields 
and such that can holonomously be transformed away, as for 
instance the 'centrifugal force'. 

10a. In fact, consider for example the galilean line-element 
in three dimensions, i.e., for <f> = const. = t/2, 

ds~ = c 2 dt 2 — dr~ — r-dd' 1 , 

taking ct, r, d as x 4 , Xi, x 2 respectively. Calculate the corres- 
ponding Christoffel symbols. Since gn= — 1, gja = — r 2 , g« = 1 , 
and all other g lK vanish, we have, for instance, the non-vanish- 
ing symbol 

i22l 
1 

But who would call it a 'component of the gravitational field '? 
This case is a particularly drastic one, for the world-geodesies 
corresponding to our line-element do represent uniform recti- 
linear motion. The appearance of non-vanishing Christoffel 
symbols is simply due to the use of polar instead of cartesian 
co-ordinates. 

In short, gravitation certainly contributes to the Chris- 
toffel symbols, but so does also a mere transformation oi 
space-coordinates, although it. has nothing whatever in 
common with 'gravitation' of the permanent or the non- 
permanent kind. This criticism does not in the least diminish 
the value of the general equations of motion (15). It is given 
here only to prevent misconceptions which have seemed 
particularly likely in the case of beginners. 

*In the terminology of the tensor calculus, to be explained later on, the 
Christoffel symbols are not the components of a tensor. 



30 Relativity and Gravitation 

10b. Let us take yet another simple example, this time 
not for the sake of criticism but because of its instructiveness. 
Consider the line-element arising' from the galilean one, 
just quoted, 

(S') ds 2 = dx' i 2 -dr' 2 -r' 2 dd' 2 , 

by the transformation 

6' = 6+00x4, x'i = Xi, r' — r, (16) 

that is to say, the line-element 

(S) ds 2 =(l- r 2 <a 2 )dx^ - dr 2 - rW - 2cor 2 d6dx i . 
In this case, taking r, 6 as Xi, x 2 respectively, the non-vanishing 
gtK are 

g 11 = - 1 , £22 = - r 2 , g 24 = - wr 2 , g u = 1 - co V. 
From these we derive, by (13), as the only surviving Chris- 
toffel symbols, 



0)% 



r22~i r24~i r44"i , 
Li_r r ' lij =0 "'±ir ar - 

Next we have, the determinant of the form (S), 

g = gn(g22gu-g2i 2 )=r 2 , 

and 

arr 2 — 1 
gii=_l 7g2 2 = ' t g24==a;>g 44 =1> 

r 2 

while all other g lK vanish. Thus we find, by (14), as the only 
non-vanishing symbols, 

12) l-2co 2 r 2 (12) (14) 

= -2cor, < > = — (l-2orr-), 



ftl 



2 r 4 — '2 



141 n „ 22) (44) • (24 



l }~*- {?}—{?} 



4 j " w '' (1 J~ '' (l j = -arr > 1 1 r = 

again seven in number. Substituting these ChristofTel 
symbols into (15), with i = l, 2, 4 (for r, 6, Xi = ct), we have 
the equations of the world-geodesies, i.e., the equations of 
motion of a free particle in the system S, 



Rotating Sysi em 



31 



r =r (d+oiXiY 

=-— (l-2co 2 r 2 )(0 + o;.-v,) 



(17) 



X4 = 4o)r.r (0 +o)X 4 ), 

where the dots stand for derivatives with respect to s. In 
virtue of the identical equation 5 = 1, i.e., 



( 1 - r 2 w 2 )x 4 2 - r 2 — r 2 6 2 - 2wr 2 dXi = 1 , 



(18) 



one, say the third of (17), should be a consequence of the 
remaining two.* Thus, the proper equations of motion in theS- 
system being the first two alone, we can use (18) to eliminate 
from them x A , and to replace d/ds by d/dt. 

In the first place, to see the approximate meaning of these 
equations of motion, consider the case of small velocities 
dr/dt, rdd/dt (as compared with c), and of small values of ur. 
[Notice that, by (16), « = a>c is an angular velocity, in its 
dimensions at least, so that ur= 6or/c is a pure number.] Thus 
ds=¥dx4 = cdt, £ 4 =f1, and the approximate equations of motion 
of a free particle in 5 are 

dr 2 dd_ 

dt 



dt 2 

d 2 6 

dt 2 



-(*+■&. 



(a) 



= — 2 



dr 

dt 



(*+-£) 



In Cartesians, 
identical with 



x = rcos6, y = rsin0, these equations are 



d 2 x 
~dF 

(Py_ 
It 2 



-2 V i o - d y 

dt 



. dx 
dt 



(b) 



The reader will recognize at once in the right hand member of 
equation (a) or in the first terms of (b) the purely radial 
centrifugal acceleration (or 'force' per unit mass), provided, 



*The verification may be left to the reader as an exercise. 



32 Relativity and Gravitation 

of course, that he is at all willing to interpret o5, in accordance 
with the transformation d' = d-\-u>t, as the angular velocity of 
the system 5 (say, plane disc) relatively to the galilean S'. 
The second terms of (b) express then the Coriolis acceleration. 

If we so desire we may, with Einstein, reckon these accelerations to the 
gravitational ones, especially if we are confined to the (rotating) system S. 
The centrifugal acceleration, at least, is radial, though away from the 
origin. The Coriolis acceleration, however, is perpendicular to the 
velocity and, therefore, generally oblique. Certainly we have in (17) a field 
of acceleration, but the only feature this has in common with a gravita- 
tional field is that all bodies placed in it will behave alike. But unlike 
gravitational fields they cannot be deduced from the distribution of matter. 
Yet Einstein would not like to have us distinguish them from gravita- 

/22( J24I 1441 
tional fields. If so, then \ , ( , ) i ( > i i ( contribute to the centrifugal, 

and < " > , \ > to the Coriolis field. But until we are told how to 

derive these non-permanent 'fields' as gravitational effects of all the 
masses of the universe turning around S* all this will be an idle question 
of pure nomenclature. We may leave it here for the present. 

In the second place, returning to the rigorous equations 
(17), consider a particle, placed (by an 5-inhabitant) at any 
point r , 8 of the disc 5 and left there, at the instant t = 0, to 
its own fate. If it is nailed down it will, of course, remain 
there for ever, being simply part of this reference system. 
But let it be a free particle from t = onwards. In short, let 
r = = 0, for / = 0. Then, by (17), we shall have, for that 
instant, 6 = so that the particle will not evince any tendency 
of moving transversally, and 



d*r . _d_ 
ds 2 dxi 



(. dr\ 5 
V dt / 



By (18), x 4 2 = (1 — r 2 co 2 ) , and since r o = 0, the last equation will 
become, rigorously, and always for / = 0, 



d 2 r .„ 

=co 2 r. 

dt 2 



*This was tried by H. Thirring but not very s uccessfully. 



Rotating S\ stem 33 

In fine, our particle will initially experience tin- familiar 

centrifugal acceleration.* It will fly off, for an V-ob-erver 
at a (straight) tangent, but from the S-standpoint at a 
spiral-shaped orbit. 

This is perhaps the clearest way of Btating the relation 
of our system S to the galilean S'. The reader need noi , 
however, think of 5 at this stage as a material rigid dia 
rotating uniformly with respect to the fixed stars, although 
a uniform rotation is just one of the possible motions of a 
relativistically rigid body (Born, Herglotz). Notwithstanding 
that 5 was called, in passing, a disc, it will be safer to treat 
it here simply as a system derived from S' by the trans- 
formation (16) with aj as constant. 

As to the orbit of a free particle relatively to S, its equation 
could be derived, not without some trouble, from the differ- 
ential equations (17). This, however, can be done much 
easier by transforming the orbit from S' to S. In fact, the 
former being a galilean system, a free particle describes in it, 
uniformly, a straight line. Its equation can be written 

r' cos 8' = r f = const., 

where r ' is the shortest distance of the straight orbit from 
the origin. Transformed by (16) the orbit in 5 will be 

12 = cos (6+5>t'), 
r 



and since v't'= V r- — r 2 , where v' is the constant S'-velocity 
of the particle, we shall have ultimately, as the orbit of a free 
particle in S, 

^ = cos [ e± %° v / ZT,~\. (19) 

r L v v r 2 _l 



*One of Einstein's most vigorous exponents, de Sitter, sees herein a 
particularly extravagant property of the rotating system. Thus in Monthly 
Notices of the Roy. Astron. Soc, vol. 77 (1916), p. 176, de Sitter says: 
'For rco<l' [and, as we saw, for any ro>] 'it is a physical impossibility for 
a material body to be at rest in the system B' [our S\. 'This shows the 
irreality of the coordinates', etc. But such is, in reality, the behaviour of 
free particles in a system rotating relatively to the stars, independently of 
any theory. 



34 Relativity and Gravitation 

which is a kind of spiral. Notice in passing that between any 
two points A, B of the disc there are two such orbits, one 
leading from A to B and the other from B to A . Thus free 
motion in 5 is not reversible. This holds also for light rays, 
for which v' in (19) is to be given the value c. Light propaga- 
tion is irreversible, and the two rays AB and BA enclose a 
certain area having the shape of a biconvex lens. But this by 
the way only. 

The example of these two systems, S' and S, was here 
treated at some length in order to acquaint the reader with 
the handling of the geodesies and the Christoffel symbols. 
At the same time, however, it may serve as a good illustration 
of the purely formal part played by the principle of general 
relativity or general covariance. In fact, although the equa- 
tions of motion of free particles have exactly the same form, 
(15) and (15) dashed, in the two systems, yet it is scarcely 
possible to imagine a more different phenomenal behaviour 
of free particles than is that in these two systems. The same 
remark applies to the light equations, gj dxj dxj = in S' 
and g uc dx l dx K = in S, exhibiting the same general form, but 
representing entirely different systems of optics; this differ- 
ence goes even so far that, while in S' all light paths are 
reversible, in S, under appropriate conditions, Brown could 
see Jones without being visible to him, though both were 
well enough illuminated. 

The purpose of these remarks is by no means to minimize 
the heuristic value of the general relativity principle, but only 
to show its purely formal nature. Notice that the case of the 
special relativity theory was altogether different; for, though 
giving privileges only to a certain class of systems, it 
claimed at least for all of them not only a formal equality, 
but an equal physical behaviour. 

In passing from S' to 5 the Lorentz contraction was, for 
the sake of simplicity, altogether disregarded. This is the 
reason why the reader was warned not to take our .S strictly 
as a rigid body rotating in S' but only as one obtained from 
S' by the simple mathematical transformation (16). Yet 
even with the said neglect the abstract 5 can at least 



Kni \ [IONS oi Monoi 

approximately stand for a rigid body, such as the <-.trtli 
plane r, parallel to its equator), endowed with a uniform 
spin relatively to the stars. 

11. Leaving these simple examples let us once more return 
to the genera] (filiations of motion of a free particle, 

in order to see what form they assume when the g iK differ bill 
little from the galilean coefficients #« and when x u .r_>, .v 3 are 
small fractions, that is to say, when the velocity of the particle 
is small compared with that of light. 

If the galilean line-element is written in Cartesians we 

have gii = g 2 2 = g33= -1, #44=1, and 

gn= - 1 + Yn, etc., g 4 4= 1+74-i, 1 / 21 x 

where all the 7 are small fractions of unity. With these 
values of the g^ we could compute the approximate values 
of the Christoffel symbols appearing in (15), and thus arrive 
at the required equations. But it is simpler to return to 
8fds = 0, the original form of (15), to reduce the element ds 
and then to develop this form afresh. 

Now, if dxi/cdt = f3 u etc., i8rH-/3 2 2 +|83 2 = j8' 2 , the line-element 
can be written 

^ 2 = </* 4 2 {l-/3 2 + 744-(7n& 2 + • • • +733/3 3 2 )+2(7 12 i8i&+ . . . 

+ 73103|8l)+2(7l4i8l+ • • . +734&)}. 

All the squares and products of the j8's are small of the second 
order. Thus, up to the third order we have 



ds = Ldx i = dx 4 Vl-p + lu + 2{p l y u + . . . +0sYm), (22) 
and the equations of motion, JSL . <:/.v 4 = 0, will be 
d 



±(dL\_BL =Qi=lt 



2,3. 

d: ' 



36 Relativity and Gravitation 



L--J_C -R) dL - 1 T 1 dyu +B djil 4- 

Jf L dx t - L L-2 dX{ dX{ —I 



Now, 

and if the squares and the products of the 7's and their 
derivatives be neglected, we can putZ,=l in the denominators. 
Thus the equations of motion will become 

d , o\ -i ^744 , a d74i , a dy i2 . a dy i3 
(7«- ft) = h • r-Pi ■ +/3 2 \~Pz 



dxi dxi dxi dxi dXi 

or, developing the first term and remembering that the y L 
differ from the g lK only by additive constants, 



d 2 Xi 

IF 


C 2 dg u , r dgq dxi a 
2 dxi L dt dt ' 




, ( ^i _ dg43 
V dx 3 dXi 



/dgu _ dgu \ 
\ dxi dXi ' 

)]■ 



+ ••• 

(23) 



These are Newton's equations of motion. The first terms 
on the right hand represent the rectangular components of 
an acceleration which is the gradient of a newtonian potential 

c 2 

= — — — g44 , 

or, vice versa, — gu plays the r61e (apart from an additive 

2 
constant) of the potential multiplied by • 

c 2 

The second terms look less familiar. But their meaning 
can be made clear at once. They represent at any rate a 
certain acceleration field which need by no means be negligible 
in comparison with the newtonian one. The contributions of 
this field to the components of acceleration are 

r dg ix dx 2 / dgu _ dg 42 \ _ dx z / dg 43 _ dgq X~\ 
L dt dt \ dx 2 dxi '■ dt V dxi dxz / -I 

or in ordinary vector language, with r=(xi, x 2 , x 3 ) and 

&4=(g41, g42, giz) 

d*v dgi XT dr 

= c — cv curl g 4 . 

dt 2 dt dt 



Equations <>\ Moi ion 

This Is manifestly the acceleration due to .1 velocity Held ^gi 
impressed upon the system of reference. It this velocity 
field is homogeneous and constanl in time, its contribution to 

acceleration is, of course, zero; bill if it is heterogeneous and 

variable, it contributes to the acceleration oi .1 free particle 
through its time rate of variation auo! through the vorticosity 

of its distribution. The simplesl case occurs when g., i- ,i 
linear function of the coordinates alone, say 

g\\ = -#2i 242= *li J?43 = , 

C C 

where w is a constant. Then c curl g 4 is a (three-) vector of 
size 2o> directed along the x 3 — axis and the last equation gives 

d-xi _ 9 - dx 2 d-x* 9 . dx } d 2 x 3 _ 

dl 2 dt ' dt 2 " dt dt 2 

which is the Coriolis acceleration corresponding to a uniform 
rotation of the system with angular velocity o> round the 

aca - axis (vectorially, with the angular velocity — . curl g 4 ). 

The reader will, perhaps, miss the centrifugal acceleration 

C) 2 r, Coriolis' faithful companion. But this (having a scalar 

potential) is inseparable from g 44 . It is included in g 44 through 

o>f 2 
the term , already familar to us from a previous example. 

c 2 

The ga just given will be found by noticing that in (5), p. 30, 
rHBdXi— (x\d Xo — x-idx\)dxi. This settles the question. 

In the more general case the spin \c. curl g 4 will not be 
constant but will vary from point to point giving rise to a more 
complicated acceleration field.* 

The approximate equations of motion (23) can now be 
written compactly, in three-dimensional vector language, 



dt 2 2 



f d l± - V * curl g, 1 . (23a) 
L dt dt J 



*I propose to call so all fields corresponding to any ds-, and to reserve 
the name of gravitational fields for those only which are 'permanent' or 
cannot be transformed away holonomously. 



38 Relativity and Gravitation" 

This equation brings at once into evidence the parts played 
by g 44 and by the three ga condensed in g 4 . Both roles may 
be equally conspicuous, and it would certainly be unjust to 

say. with Einstein, that it is only gu which survives in this 
first approximation. 

Einstein (loc. cit., p. 817), in deriving the approximate newtonian 
equations from the rigorous ones, no doubt, through a too hasty computa- 
tion of the Christoffel symbols, dropped altogether the second terms of 
our equations (23). And his 'slip' crept into the writings of de Sitter, 
Weyl and others. Einstein exclaims even (ibid.) in genuine surprise: 

EdrX{ c 2 dgu ~~1 
= — — is that onlv 
df- 2 dxt _ 

the component gu of the fundamental tensor determines by itself, in a 
first approximation, the motion of a material particle'. 

We shall return to these approximate equations of motion 
later on, after having set up Einstein's gravitational field- 
equations. 



CHAPTER III. 

Elements of Tensor Algebra and Analysis. 



12. In order to be able to construct generally covariant 
laws or equations, such as Kinstein's field-equations which 
will complete the fundamental part of his theory, some 

elementary notions of the Tensor Calculus are required. 
These I shall now proceed to give, without stopping to sketch 
the history of the origin and the growth of this powerful 
method of multidimensional analysis, which the reader will 
find in the preface to Ricci and Levi-Civita's paper on the 
Absolute Differential Calculus,* as the said branch of mathe- 
matics is called by these authors. 

The relations and properties which are now to occupy 
our attention hold for a manifold of any number of dimensions. 
But, if not otherwise stated, we shall have in mind our four- 
dimensional world or space-time. 

A world-point is given by four gaussian coordina; 
which, in general, are mere numbers or labels. As such they 
need not, as in the most familiar treatment, stand for such 
things as lengths or distances, or angles. By calling them 
'labels' we do not mean, of course, that tetrads of numbers 
are being haphazardly, disorderly, attached to various events 

*G. Ricci and T. Levi-Civita, Methodes de calcul differentiel absolu el 
leurs application, Mathem. Annalen, vol. 54 (1900), pp. 125-201. A con- 
densed account of this paper is given in J. E. Wright's Invariants of Quad- 
ratic Differential Forms, Cambridge Tracts, No. 9 (1908). Perhaps the 
easiest presentation of all that is required for relativistic applications is 
given in the second part (B.) of Einstein's own paper, loc. ci!., essentially 
reproduced in chap. Ill of A. S. Eddington's Report, Phys. Soc. London, 
1918. Th subject is treated on original and very attractive lines by H. 
Weyl in Rauin, Zeit, Materie (Springer, Berlin), 3rd ed., 1920. For geo- 
metrical applications the first volume of L. Bianchi's Lesion; di Geon: 
Differenzialc (.Spoerri, Pisa), 2 ed., 1902, can be most warmly recommended. 

39 



40 Relativity and Gravitation 

(world-points), but we assume that Xi = 7, say, is a label 
attached to a whole connected three-dimensional continuum 
of world-points, and similarly for all other (real) numerical 
values of x%. Likewise for the remaining coordinates, so that 
every world-point appears as the intersection of, or element 
common to, some four hypersurfaces of three dimensions. 
Manifestly, the use of such coordinates does not presuppose 
any idea of measurement. Again, in this abstract treatment 
of tensors as certain entities in the manifold, the question 
whether any one of the coordinates or its differential is space- 
like or time-like, is of no interest. It becomes relevant only 
when we come to apply these concepts to physical problems. 
13. Such being the nature of the x L , pass from these to 
any other coordinates x/, through any holonomous transforma- 
tion whatever, satisfying only the conditions of continuity, etc., 
as stated in chapter I. Then, as in (8a), the differentials dx„ 
i.e., the coordinates of a world-point Q, a neighbour of 
P(x L ), with P as origin, are transformed into 

7 . dxj 7 dxj 7 , dx L ' 7 

dx l — — - dx K = dx\ + dxi -y . . . . 

dx K dx\ dx% 

That is to say, the coordinates of Q with P as origin, are given, 
in the new system, by these linear homogeneous transforma- 
tions of the old relative coordinates of the pair of points, 
with coefficients, dx//dx K , which are some given functions of 
the position of P. Such an ordered point-pair, PQ, or the 
corresponding array of the dx L , is called a vector, in our case 
a four-vector or world-vector. From a more general standpoint 
to be explained presently its name is: a contravariant tensor 
of rank one. 

Now, as in special relativity every tetrad which is trans- 
formed as the cartesian x, y, z and ct (i.e., by the very special, 
linear, Lorentz transformation), so here the tetrad of infini- 
tesimals dx L is made the prototype of all (contravariant) 
vectors. In other words, every tetrad of magnitudes A 1 
which are transformed by the same rule as the dx„ i.e., 

dx' 

A' l =°^-A K , (24) 

dx K 



Covariani \'i < ro] 41 

is called a conlravariant vector or a contravariant tensor of rank 
one, and A\ A' l i, etc., are called its components. (The upper 
position of the suffixes was proposed by Ricci and Levi-Civita 
and accepted by all authors. To be consequent one would 
have to write also dx 1 , as in fact is done by Weyl. But, for 
the sake of typographical convenience, an exception is being 
made for this prototype of all contravariant vectors.) It is 
scarcely necessary to say that, unlike the Cartesians in special 
relativity, the coordinates x t themselves do not form a vector; 
only their differentials do. In short, there are, in general, no 
finite position-vectors, but only differential ones. This, how- 
ever, does not exclude the possibility of other finite vectors A 1 . 

It is of particular importance to notice the linearity and 
homogeneity of the transformation formula (24) which will 
reappear in the case of all other tensors. The all-important 
consequence of this property is that if all components of a 
vector vanish in one system, they will vanish also in all other 
systems of coordinates. More briefly, if a vector A K vanishes 
in one system it will vanish also in any other system. Thus 
A K = will be a generally 'covariant' or, technically, contra- 
variant law. This, of course, does not prejudice the question 
whether Nature is going to obey it. 

Manifestly, if A" and B" are two contravariant vectors, so 
also are A K +B K and A K -B K . 

As dx, served as the standard of contravariant vectors, so 
do the operators (differentiators) 

dx, 

serve as a prototype of another kind of vectors. We have, 
evidently, 

dx 

dx/ 

and every tetrad of magnitudes B t which are transformed 
according to this rule, 

B/=p!LB K1 (25) 

dx, 



42 Relativity and Gravitation 

is called a covariant vector or tensor of rank one. In com- 
parison with (24), notice that the suffix of B' coincides with 
the lower (instead of upper) suffix in the coefficients. Although 
the prototype of these vectors consists of differentiators, the 
components B l of a covariant vector need not be operators, but 
may be magnitudes in the ordinary sense of the word. As 
in the previous case, B K = is a generally covariant equation 
or rather set of equations. And if B K and C K be two covariant 
vectors, so also are B K ±C K . Needless to say that A K -\-B K is 
neither a covariant nor a contravariant vector. In fact, it has 
no meaning if the system is not specified. 

14. But, while the sum of a covariant and a contravariant 
vector is from the present point of view of no interest, the 
combination of their components 

A l B=A 1 B 1 -\-A^B 2 +A 3 B 3 +A i B i , 

which is called the inner or scalar product, has a very remark- 
able property. It is invariant with respect to any transfor- 
mations of the coordinates. In fact, by (24) and (25), 

A "B t ' = ^A k ^±B x = (— x ^') A K B X . 

dx K dxj \dx/ dx K s 

But the Xi, x 2 , etc., being mutually independent, the bracketed 
expression (to be summed over all t) vanishes for all k=f X and 
equals 1 for /c = X. Whence, 

A n B! = A K B K = A l B„ (26) 

which was to be proved. 

Any invariant, S=S r , is also called a scalar or a tensor of 
rank zero, since, in a manifold of n dimensions, it has n° com- 
ponents, i.e. but one component. Similarly, a vector or 
tensor of rank one, has n l = n, in our case four, components. 
The question whether a scalar is a contravariant or a covariant 
tensor is idle. For it transforms into itself. 

Vice versa, it can easily be proved that if B K be four 
(generally, n) magnitudes such that A K B K is invariant for 
any contravariant A K , then B K is a covariant vector. And 



Second k.wk Ti 13 

the same thine, is iruc if ' co variant' and 'contravariant* 

he exchanged with one another. 

The product of a vector by a scalar is, obviously, again a 

vector of the same kind, and any number "I v< . tora oi 
same kind multiplied by scalars and added tog«il 
again a vector of the same kind. Finally, notice that AJB, 
and A K B K are not invariant, and thus are no tensors at all. 

15. As we just saw, the inner multiplication of a covariant 
and a contravariant vector degrades the rank of both factor- 
giving a tensor of rank zero, a single component. Consider, 
on the other hand, what is known as the outer product of two 
vectors, of the same or of opposite kinds, i.e., A,B K , or A l B", 
or A t B K . The suffixes being here different, no summation i> 
understood, so that each of these symbols stands for 4 2 =1G 
(generally w 2 ) components. Let us take A L B K first, which is a 
short symbol for the array 

AxB x ArB 2 . . . 
A 2 B y A2B2 . . . 



of sixteen magnitudes. Denote them by M^ respectively. 
Their law of transformation is, by (25), 

M' m -p-pLM+ (27) 

dx L dx K 

Every array of n 2 magnitudes N lK (whether obtained by the 
outer multiplication of two covariant vectors or in any other 
way) which is transformed by the rule (27) is called a covariant 
tensor of rank two. It manifestly has again the property of 
vanishing in all systems, if it vanishes in one of them. In a 
four-manifold N lK consists of 16 components. 

In general N lK + N Kl . If, in particular, N lK = N Kl the tensor 
is called symmetrical. 

An example of such a tensor we had in g lK , called the 

fundamental tensor; cf. formula (9). Notice, however, that 

the tensor property of g lK followed from the invariance of ds' : 

which fixed the metrical properties of the world, whereas all 

our present considerations are entirely independent of the 
—4 



44 Relativity and Gravitation 

metrics of the manifold, and it is preferable to abstain from 
using them at this stage. Such properties as are impressed 
upon the general tensors by the metrics of the world will be 
treated in later sections. 

In the meantime let us continue the non-metrical theory of 
tensors. 

The symmetrical tensor N lK consists in general of |w(w+l), 
and for w = 4, of ten different components. It can be easily 
proved, by (27), that its symmetry is an invariant property, 
i.e., that if N lK =N Kl in one system, we have also N f lK =N' Kl in 
any other system. A covariant symmetrical tensor of rank 
two can be constructed at once from a covariant vector, to 
wit by forming its outer self-product, A llv = A ll A v = A v A lli = 
A Vfi . 

If N lK = —N KL , for all i, k, we have an antisymmetrical (or 
skew) tensor. Since N KK =—N KK means N KK = 0, a whole 
diagonal of components vanish, and thus only ^w(w-fT) — n = 
%n(n—l) non-vanishing and independent components are 
left, the surviving ones being oppositely equal in pairs. 
Thus an antisymmetric tensor in a four- world consists of 
six independent components, and is therefore called a six- 
vector, in the present case a covariant six-vector. With such 
six-vectors the reader is already acquainted from the special 
relativistic treatment of the electromagnetic field. We shall 
see them at work in a similar duty in general relativity 
later on. 

As the symmetry so also the antisymmetry is an invariant 
property, i.e., iV tK = — N Kt is transformed into N' tK = — N' KL . 

Any tensor N tK can be split at once into a symmetrical 
and an antisymmetrical one. For we have identically 

N lK =i(N lK +N Kl )+UN tK -N Kl ), 

and the first term represents a symmetrical, the second an 
antisymmetrical tensor. 

Similarly to (27), and starting from the special tensor 
A l B K , any array of n 2 magnitudes which are transformed by 

the rule N ,iK = -^ bx -L. N aP (27a) 

dx a dxp 



Mixed Tensors 15 

is called a conlravarianl tensor of rank two. If N" = A*", it is 
a symmetrical, and if N"= —A 7 " 1 , an antisymmetrical tei 
(A tensor A 7 "' need not be the product of two contravarianl 
vectors.) 

Lastly (starting from AJ¥), any array of n- magnitudes 
A 7 ," which are transformed by the mixed rule 

N , K = dx^ dxj_ N * (276) 

dx/ dx a 

is called a mixed tensor of rank two, covariant with respect to 
its lower suffix t, and contravariant with respect to its upper 
suffix or index k* Special cases of symmetry and anti- 
symmetry as before. A new feature, however, offered by 
the mixed tensor is this. With any A 7 " make t = /c, getting 
N K K and, by the usual convention, sum over all k. In other 
words add up all the components of the chief diagonal (slanting 
down from left to right) of the mixed tensor. The result will 
be a single magnitude. Now, the important thing is that 
this magnitude is a general invariant. In fact, by (27b), 



\dx \. dx n / 



but (as mentioned before) the bracketed expression is zero 
for all a?±fi and one for a = fi. Thus 

N'l=Nl=Nl, 

which proves the proposition. 

Thus, equalling the upper and the lower index and 
summing over it degrades the mixed tensor by two rank- 
giving, in the present case, a tensor of rank zero or an 
invariant (scalar). In other words, 

n:=n 



*It seems inappropriate to call 'surrix' (from sub, under) an upper mark 
or sign. I propose .therefore, to call such signs by the more general name 
■index. Since all English writing authors accepted the ' three-;;; dex symbols ' 
and the 'four-index symbols' (of Christoffel and Riemann), they will per- 
haps not object to calling t, k indices. 



46 Relativity and Gravitation 

is an invariant of the tensor N K ,. We shall see presently that 
this procedure of equalling an upper to a lower index, called 
contraction (German ' Verjiingung') can be applied, with equal 
success, to a mixed tensor of any rank whatever. Notice, 
however, that this process is not applicable in the case of 
(purely) covariant or contravariant tensors. Thus, for 
instance, M KK = M n + M22 + ... is not invariant, as a glance 
on (27) will suffice to show. In short, the diagonal sum of 
M lK has no intrinsic meaning. Similarly, in the case of a 
four- vector, say, Ai~\- . . . -\-A± is not an invariant. 

16. The next step, leading to tensors of rank three, and 
so on, is obvious. Generally, any system of w r (in our world, 4 r ) 
magnitudes iV*. ';;, with r\ lower and rz upper indices, which 
are transformed by the rule 

/■pja0...\r _ vXj dXfc OX a OXp /y? 6 --- (28") 

dxj dx K ' dx a dx b 

is called a mixed tensor of rank r = ri-\-r 2 , covariant with respect 
to its r± lower, and contravariant with respect to its r 2 upper 
indices. If all the components of such a tensor vanish in one 
system they will also vanish in any other system of coordinates. 
Any tensor, therefore, can be used for writing down generally 
covariant laws.* In particular, if ri = 0, the tensor (28) is con- 
travariant, of rank r 2 ; and if rz = 0, covariant of rank n. The 
sum of any number of tensors of the same rank and kind, 
each multiplied by any scalar, is again a tensor of the same 
rank and kind, the numbers n, r 2 retaining their significance. 

17. Contraction. This process, already illustrated on the 
simplest example, can now be generally explained. 

Let a be any upper and t, any lower indexf of a mixed 
tensor of any rank r whatever. Put a = t and sum over a. 
Then the result will be a tensor of rank r — 2, with r\ — \ covari- 
ant and 7"2— 1 contravariant indices. 



*In the less technical sense of the word. 

•jThe place of a among the upper, and of t among the lower indices is 
irrelevant. 



Contraction of Tensob I. 

The proof follows at once from (28). For the process gives 
us in the coefficients of transformation a term 

dx a dxj 

dx,,' dXi 

which vanishes for all a^i and equals one for a = i, thus 
reducing (28) to 

(flf;:;)'- **L ^L ....Afc 



dx/ d.r*. 



Ar... . 



which proves the statement. 

This process of contraction can obviously be applied again 
and again, degrading the tensor each time by two ranks until 
there will be no upper or no lower indices left. In fine, tin- 
mixed tensor can be degraded until it becomes purely covariant 
or purely contravariant or (if ri = r 2 ) until it is reduced to a 
scalar or invariant. 

Thus, for example, the tensor A*j* x of rank five gives rise to 

A aK \ = ^n\ + A^ K \ + • • • l 

which is denoted by A% x , and this tensor of rank three gives 
rise to 

4* = ^x 

which is a (covariant) tensor of rank one or a vector. 

Again (as an example of r\ = r 2 ) . the tensor A*j* of rank four 
gives by contraction A% , and this tensor of rank two gives 

a: = a, 

a scalar. We may as well write at once A™ = A, the meaning 
and the value of A being the same as before. This final 
invariant may be considered as a property of the original 
tensor A"? . 

In general every such half-and-half tensor (fi=rs) will have 
the final scalar (A) as its intrinsic* invariant. And, as far as 
I can see, this is its only intrinsic invariant. 

*i.e. an invariant of its own, independent of any extraneous form such 
as ds 2 (or any auxiliary tensor, such as g lK ) determining the metrics of the 
manifold. 



48 Relativity and Gravitation 

On the other hand a purely covariant or contravariant 
tensor or an unequally mixed one (fi?^) cannot be contracted 
to an invariant. It seems that it has no intrinsic invariant 
at all, that is to say, that there are no processes which 
would lead to an invariant combination of the components 
of the original tensor itself (without using other tensors). 

18. The inner multiplication, already mentioned in con- 
nection with vectors, can now be considered as an outer 
multiplication followed by a contraction. 

Consider two tensors, generally mixed, one of rank r = ri~\- 
r 2 , the other of rank 5 = 5i+52- Combine (by ordinary multi- 
plication) each of the n r components of the former with each 
of the n 3 components of the latter. The n r+s magnitudes thus 
obtained will be the components of a tensor of rank r+s 
with fi+Si covariant and r 2 +s 2 contravariant indices. That 
the entity thus arising is a tensor follows at once from (28). 

Thus the outer product of two vectors is a tensor of rank 
two, A t B K = M tK , A L B K = M K L . Similarly A afi B lK is a covariant 
tensor of rank four, M a p lK , and A aP By = N L ^ y is a mixed 
tensor of rank five, and so on. 

The outer multiplication combined with contraction 
(when there are indices to contract) gives the inner product. 
Thus the inner product of A L and B K is 

A K B K =M K K = M, 

an invariant.* The inner product of A K and B a $ is their outer 
product M K a p degraded by contraction, i.e., M% = M a , a covari- 
ant vector. The inner product of A a p and B lK is their outer 
product A a pB lK = M l *p degraded (to the utmost) by two 
contractions, 

m k :=m, 

i.e., a scalar or invariant. Vice versa, if A a p be any array of 
n 2 magnitudes such that A aP B lK is an invariant for any con- 
travariant B lK , then A a $ is a covariant tensor of rank two. 
This criterion of tensor character, already mentioned in con- 
nection with A L B K , can be easily proved by writing down the 

There is no inner product of A L , B K . 



Tensor Differentiation 19 

transformation formula of the given factor (tensor). And it 

can be extended to any rank and kind, no matter whether 
the inner product is a scalar or a tensor of any rank higher 
than zero. 

As we already know, the differential operators D t = • 
have the character of the components of a covariant tensor 
of rank one. Therefore, the 'product' of this tensor into a 
scalar or scalar-field f=f(xi, x 2 ■ ■ ■), that is to say, the result 
of operating with Dl upon/, will again be a covariant tensor 
of rank one or a covariant vector, 

^- = A u . (29) 

Ox, 

But we cannot go further than that. That is to say, an iterated 
application of the operation D K does not give a tensor. Thus 
d' 2 f/dx L dx K is not a tensor. Nor do, in the more general case 
of any vector B u the n 2 derivatives D K B t = dB L fdx t constitu te a 
tensor. The different behaviour of D K B l and of products of 
magnitude-tensors lies herein that the operational tensor D K 
acts also on the coefficients dx K /dx,_' of the transformation 
formula of B t . In fact, we have 

DW-*±D m (23LBd, 

dx K ' V dx t ' / 

and* this is not the same thing as —2 —?- D.B 8 . The same 

dxj d\\' 

remark applies, a fortiori, to higher derivatives of scalars and 

of tensors of any rank. 

In fine, the only tensor derivable by simple differentiation, 
unaided by other auxiliaries (cf. infra), is the covariant vector 
(29) yielded by a scalar. The vector or vector-field df d.\\ is 
called the gradient of/. In the case of space-time it consists 
of four components. 



*Unless the coordinate transformations are linear as in the spoci.il 
relativity theory. 



50 Relativity and Gravitation 

19. Tensor properties in a metrical manifold. Having 
sufficiently acquainted ourselves with the properties of tensors 
in themselves, let us now consider them in relation to the 
fundamental quadratic form ds t = g lK dx,dx K which converts 
the hitherto amorphous world into a metrical or riemannian* 
manifold. 

It is of the utmost importance to grasp well this distinction 
between a riemannian and a non-metrical manifold and to 
understand the true role of ds 2 in converting the latter into the 
former. 

Let us place ourselves yet for a while upon the non-metrical 
standpoint. Of all the tensors described in the preceding 
sections let us confine our attention upon the prototype of all 
(contravariant) vectors, the infinitesimal position-vector dx t . 
Any such vector represents ultimately but an ordered pair of 
points, 0(x L ) the origin, and A(x l ~\-dx l ) the end-point of the 
vector. Imagine a whole bundle of such infinitesimal vectors 
OA, OB, OC, etc., all emerging from the same world-point 
O as origin. Now, from the non-metrical point of view, all 
these vectors have (apart from their origin) nothing in common 
with one another. That is to say, if two of them, say OA 
and OB, are at all distinct from one another, and if their 
components dx L do not happen to be proportional to one 
another (in which case we can say that the vectors have a 
common 'direction'), there is in either of them nothing, no 
property, with respect to which they could be compared. They 
are, as it were, perfect strangers to one another. Similarly, if 
we call 'angle' a vector-pair a — OA, OB, there is nothing to 
base upon a comparison of two non-overlapping covertical 
angles a and (3 = OC, OD. In short, neither vectors nor angles 
(or other derived entities) have 'sizes'. There is, in fact, in 
the manifold itself nothing which could fix the mere meaning 
of such a concept. Of two vectors OA, OB nothing more can 



*The name 'riemannian' manifold or w-space is being often used in this 
connection in view of the historical fact that Riemann was the first to base 
the general geometry of an rc-space upon its line-element given by such a 
differential form, although Gauss was his great predecessor in the case of 
surface theory. 



The Line-Elemi ~>\ 

be said than thai they arc either identical (or co-directional, 
collinear) with or distincl from one another. The origin 
being the same, 11 the points A , B are either identical or dis- 
tinct, and no other significant statement ran be made ab 

their relation. 

But while there is nothing in the manifold itself to base a 
comparison of distinct infinitesimal vectors upon, we an 
liberty to provide for it at our will if we so desire. This ifl 
done by introducing a standard or fundamental entity such 
as the quadratic form called the line-element. In other 
words, we surround the world-point 0{x t ) by a hypersurface, 
a three-dimensional (generally an n — 1 dimensional) quadric 
and declare all vectors emerging from and ending in any 
point P {x L -\-dx) of this surface to be equal in size or in 
absolute value, or in 'length', the usual name in the case of 
our three-space. It is precisely this metrical surfacef which 
is expressed by 

g lK dx t dx K = ds 2 = const., 

the numerical value of ds being the 'size' common to all these 
infinitesimal vectors or point-pairs. J The part played by 
this quadratic form is essentially the same as that of Cayley's 
'absolute' or standard quadric (a real quadric leading to 
lobatchevskyan or hyperbolic, an imaginary quadric leading 
to elliptic, and the intermediate degenerate quadric leading 
to euclidean geometry), the only important difference being 
that Riemann's treatment is much more general. It covers 



*We have limited the discussion to coinitial vectors solely for the sake of 
simplicity. All our remarks apply a fortiori to distant, non-coinitul 
bundles of vectors. 

tThe German geometers call it Eichflciche. 

Jin Riemann's own treatment this role of the fundamental form im- 
pressed upon the manifold extends into distance, over all the manifold. 
That is to say, if 0'(y ) be any other point and if a quadric g _dy dy = const, 
be drawn around it with the same value of the constant as before, all the 
vectors of the bundle O' terminating upon this quadric are again said to 
have the same size as those of the bundle 0. In this respect a somewhat 
more general standpoint was recently proposed by Weyl, in connection 
with his ideas on electromagnctism. 



52 Relativity and Gravitation 

all metrical spaces (in technical language, of variable and 
anisotropic curvature), whereas Cayley's device gives us 
only a space of constant isotropic curvature, negative, zero, 
or positive. This fully corresponds to his starting point, 
which was that of projective geometry. Yet, and this is of 
particular interest in the present connection, Cayley recog- 
nized thoroughly the true role of all such standard entities. 
In fact, he tells us plainly that geometrical figures have no 
metrical properties in themselves. Their metrical properties 
such as those of the foci of a conic, etc., arise only by relating 
them to other figures, as the 'absolute' conic in the plane, or 
quadric in three-space. 

The kind of metrics thus impressed upon a continuous 
manifold being essentially arbitrary, the utility of the metrical 
manifold thus obtained will, of course, from the physicist's 
standpoint, depend upon the interpretation which is given to 
the said 'size' of a position-vector, and to special lines of 
that metrical manifold, such as the geodesies, in terms of 
measuring rods, clocks, moving particles or light phenomena, 
and so on. 

But without dwelling here any further upon such questions 
of a concrete representation let us turn to consider the purely 
mathematical consequences of the introduction of g tK dx l dx K 
as a fundamental differential form fixing the metrics of the 
manifold. 

20. As in Cayley's case the geometrical figures in relation 
to his 'absolute', so here the tensors acquire some new pro- 
perties in relation to the fundamental form or better, to its 
coefficients g lK . In fact, what determines the form are these 
coefficients, and we may look upon the matter in the following 
way. 

Instead of declaring the fundamental quadratic form at the 
outset as an invariant, let us better say that the symmetrical 
array of 16 (generally ri 2 ) magnitudes g lK is being introduced 
as a. fundamental tensor, symmetrical, of rank two and of the 
covariant kind, as defined in the preceding sections. 

Combined with this fundamental tensor all other tensors 
of the previously amorphous manifold will acquire some 



Metrical Properth 

mcv. properties. These and only these will now l»<- their 
metrical properties. 

To begin with the prototype of contravarianl ve< tors, the 
infinitesimal vector dx t has had thus far no invariant of his 
own. Hut it will acquire one with the aid of the fundamental 
tensor. In fact, dx l being contravariant, denote it for the 
moment by X 1 . Form the outer product 

which will be a mixed tensor A '^ . Contract it with respect 
to i, a, getting A\p=Ap. Contract this again. Then the 
result will be A K K = A, a scalar or invariant. Or perform both 
contractions at once, and write ds 2 for A, returning to the 
original notation, thus 

g lK dx L dx K zEzds-= invariant. 

In short, the inner product of the tensor dx a dx$ into the funda- 
mental tensor g lK is an invariant. There is no objection to 
calling it the invariant oj dx t as a short name for its metrical 
or associated invariant. Thus, thanks to g u , the vector dx, 
has acquired an invariant. And it can now be compared 
through it with other vectors, no matter what their com- 
ponents. The value of ds 2 may be called the norm, and the 
absolute value of v ±ds 2 the size of the vector dx\ . Thus 
we can speak of two vectors dx t and dy t being equal in size, 
or one having twice the size of the other, and so on. In 
application to the four- world, a vector dx t of no size will be 
a light vector, a vector of negative norm a space-like, and 
one of positive norm a time-like vector. 

Similarly, any other contravariant vector A' will have the 
metrical invariant 

gut A'A K =A\ say.* (30) 



*Of course, even in the amorphous manifold an invariant could be built 
up from A 1 by the aid of any covariant tensor N lK , but the choice of W M 
being entirely free, such an invariant would not have a fixed value. We 
fix it by introducing once for all a special tensor g lK to serve for all other 
tensors. 



54 Relativity axd Gravitation 

In much the same way. if B L be any covariant vector, we 
shall have in 

g^B.B^B- (30a) 

an invariant, the norm of B L . 

From a more general point of view we may call -4-. in (30). 
the tensor, of rank zero, metrically associated to .4', similarly, 
in (30a), B- to B L . 

Moreover, we can easily construct associated tensors of a 
rank other than zero, and differing also in kind from the 
original tensor. Thus, to dwell still upon vectors, 

g^A' = A t (31) 

will be the covariant vector metrically associated with the 
contra variant vector A'. We may call A, shortly the conjugate 
of A\ Similarly, starting from a covariant vector A tt we 
shall have the contravariant vector 

g a A K = A' (31a) 

conjugate to -4, . 

Two questions naturally suggest themselves: Will the con- 
jugate of the conjugate be the original vector? Have two 
conjugate vectors the same size or the same norm? 

In order to answer these questions as well as for the sake 
of what will follow, let us first note a simple property of the 
tensors g^ and g 1 * . By definition, chap. II. g" is the minor 
of the determinant g = g„ . corresponding to its i, K-th element, 
divided by g itself. But g is equal to the sum of the produce 
of the elements of its first column, say, into the corresponding 
minors, i.e., g = g a igg ai . whence gaig° 1= l- Similarly for any 
other column (or row). Thus, underlining the index over 
which an expression is not to be summed. 

This is valid for even.- v. Thus g^g"", summed over both 
indices, has the value 4 for our world, and n for an ?i-fold. 
Again, taking two different columns (or rows) of g. we shall 
easilv prove that 



( onji GATE 'I i NSOBS 56 

Both properties can be united in formula 

«..«•' = *! = «!. 

where 6„ is the conventional symbol for 1 or according as 
a = /3 or aj^/3. This symbol is itself a mixed tensor. 

We are now able to answer our two questions. First, the 
conjugate of the conjugate of the vector A t is, by the definitions 
(31), (31a), 

n g" A = g l A = 6' - 1 = 1 

i.e., the original vector. Similarly if we started with A*. 
Thus, the conjugate of the conjugate is the original vector. 

Second, if A 1 be the conjugate of A t we have for the norm 
of the former vector, by (30) and (31a). 

g^A'A^g^g" A mj f'A fi =8 a K A^'Ap = £'A fi A m . 

Thus any two conjugate vectors have equal norms. 

The norm of A t and of A' can also be written A t A l , for 
this is again equal to g^ A l A K . Thus, for instance, if d^ be the 
conjugate of the contravariant vector dx u their common 
norm or the squared line-element can be written 

ds- = d.\\d^. (33) 

21. In much the same way we can treat the metrical 
properties of tensors of any higher rank. To explain the 
method it will be enough to take up in some detail the second 
rank tensor A u . Its conjugate or supplement (Erganzung) 
will be the contravariant tensor defined by 

ffA^-A' , or also g„a«4*-it« . (34) 

The tensor g** itself is easily proved to be the supplement of 
the tensor g lK . 

The scalar or invariant of A^ will be 

CA m -Al-A. (35) 

A single contraction of g" A a& will give 

o" A =.4 l 
a mixed tensor metrically associated with the covariant A~. 



56 Relativity and Gravitation 

The supplement of the supplement (or the conjugate of the 
conjugate) is again the original tensor, for 



g ai getA" = g ac g 0K g> 1 g SK A yi = dl 8 5 eA y& =A 



yb 



The tensors A lK and A" have the same scalar A, (35). In fact, 
the scalar of A lK is 

U A"" =&, C 2^.* = E t A aP = i K Az=A. 

Since g? v A MV is an invariant, B^ =g lK g?" A liV is again a tensor; 
Einstein calls it the reduced tensor belonging to A^,. 

Notice that neither a covariant nor a contra variant tensor has an 
invariant independent of the metrical tensor; only a mixed tensor, B K 
has such an invariant, to wit B = B K . This is a privilege of mixed tensors of 
even rank with ri=r 2 , and of these tensors only. 

The investigation of other metrical properties of tensors 
of the second and higher ranks may be left to the reader. 
Exercises of such a kind will soon make him familiar with this 
broad and powerful algorithm. 

22. Angle and volume. Consider an) 7 two coinitial in- 
finitesimal vectors dx L , dy L . These are contra variant vectors. 
Therefore, as we already know, the inner product 

o dx dv 

Sue """*i "'.»« 

will be an invariant. It will remain invariant when divided 
by the sizes of both vectors. By an obvious generalisation of 
the familiar cosine formula this invariant is used to define 
the angle e made by the two vectors, thus 

g IK dx t dy K . . 

cos e = , (ooj 

ds do- 
where ds 2 = g LK dx t dx K , da 2 = g LK dy t dy K , The two vectors are 
said to be orthogonal or perpendicular upon one another if 

g LK dx t dy K = . 

Generally, the angle between any two vectors A\ B\ whose 
norms as defined by (30) are A 2 and B 2 , will be determined by 



i.k and Volume 






cos t = 



g« A % BT 
AB 






and the vectors will be orthogonal if g„ i4\4*=0. Similarly 

for covariant vectors, with the only difference th.a ■■ 
replaced by g lK . Let A t , B K be the conjugates of A*, B l ; then 

g 1 " ^B.-g" &. g^A a B" = 5l g fiK A a B fi = g afi A a B", 

and since -<4 t , B l have the same norms as A 1 , B\ we see that 
the angle between the conjugates is the same as between the 
original vectors. 

The integral fdx\dx% ■ ■ • dx n extended over a domain of 
the manifold is, by a well-known theorem, transformed into 



fJdxi dx% . ■ ■ dx n \ where / is the Jacobian 



a.v/ 



, as in (7). 



On the other hand the determinant g of the fundamental 
tensor (called also the discriminant of the fundamental 
quadratic form) is transformed into 



dx a 



dx fi 
dxj 



dXp 

~dx7 



the last step being based on the multiplication rule of deter- 
minants. Thus 

a' = J*o, 



Consequently, the integral 

fVg dxidxi 



dx, 



(38) 



(39) 



is a scalar or an invariant of the n— dimensional domain of 
integration. 

In the case of the four-dimensional world the determinant 
g is always negative.* Thus the invariant expression 

*In a galilean domain and in Cartesians g= — 1, by (lb), p. 6. Bj 
therefore, it is also negative, always for a galilean domain, in any other 
system of coordinates derived from the Cartesians by a holonomous t: 
formation. Now, although a non-galilean domain cannot be made galilean 
by a holonomous transformation, yet we know that in all practical C 
the g lK differ but very little from the galilean coefficients. Thus f will also 
in general be negative. 



58 Relativity and Gravitation 



dQ, = v — g dx\ dxt d%z dxi (40) 

will be real. This is taken as 'the local measure' of the size 
or volume of an infinitesimal world-domain. For in the local 
(cartesian) coordinates u lt for which g= — 1, this expression 
becomes du\du^duzdu^ = cdtdxdydz. The latter product is 
called by Einstein 'the natural' volume-element. Apart 
from names, the important thing to notice is the general 
invariance of the expression (40) as such or when integrated 
over any world-domain. 

Consider any sub-domain of the world, of three, two or one dimension. 
This can be represented by expressing the x t as functions of three, two or 
one parameter respectively. The differentials dx L will be homogeneous 
linear functions of the differentials dp a of these independent parameters. 
Thus the line-element within the sub-domain will be of the form 

ds 2 = /fo0 dp a dpp , h a p = h@ a , 

and the sub-domain, therefore, will again be a metrical manifold (a three- 
space, surface or line) in Riemann's sense of the word, and if /z = |Zt a /sl» 

d£L=V h dpidpz . . . 



will (apart perhaps from a factor V — 1) again be an invariant measure of 
an element (volume, area, length) of the sub-domain. 

Thus, in the case of a one-dimensional sub-domain or line, 

j dx L 

dx^ = dp, 

dp 

and ds 2 =g lK r-—^- — — dp 2 = hu dp 2 , say. 

dp dp 

In this case h = hn and, therefore, 

dtt=V~h u dp, 
which is ds itself, as it should be. 

For a two-dimensional sub-domain or surface we have 

ds 2 = hn dpi 2 +2hu dpi dp2+hi2 dpi 2 , 

dx, dx K 

where h a b=guc ~T— "TT - • 

opa upb 
Thus, 

where 



—4 



dtt=V h dpidp 2 , 

3*i dx K dx L d-\ _ / dx t dx K y 

dpi dpi tK dp2 dpi \ lK dpi dp2 / 



< 0VARIAN1 I >him \ i|\ I 

23. Differentiation based on metrics. We have already 
(p. 49) thai if/ be a scalar or invariant, df/dx„ the gradienl 
of /', is a covariant vector. This is independent of the metric - 
of the manifold. But, as was then pointed out, the iterated 

application of the operation d t dx, would not lead to 
nor would its application to a vector A, or another tensor 
yield by itself, unaided by auxiliaries such as g u , a tensor. 
But the introduction of the metrical tensor opens in this 
respect new and important possibilities. 

It was remarked by Christoffel as long ago as INfi'i thai 
if A t be a covariant tensor, so is 



J r:K 



ox k 



namely covariant, of rank two. Similarly if B lK be a covariant 
tensor of rank two, 

** _ "5T ~ l a/ B "-\ a i B - ' 42 ' 

is again a covariant tensor of rank three; similarly 



dx x I « J ( K I 



^ = _^_ _ .a b;+ ^ . ^ (42fl) 



is a mixed tensor of rank three, and so on. But it will be 
enough to consider here at some length the first case (41) 
only, especially as the other cases can be derived from it. 
The operation indicated in (41) is called covariant differentia- 
tion, and its result A M the covariant derivative or the expansion 
(Rrweiterung) of A L . 

1 1 B' be a contra variant vector, 



B" =e 



is a contra variant tensor of rank two, the cotitravariant derivative of B. 
But for our purposes it will suffice to consider only the covariant 
differentiation. 

That (41) represents a covariant tensor can be proved in 
a variety of ways. The most instructive of these is perhaps 



60 Relativity and Gravitation 

that given by Einstein, since it makes immediate use of the 
equations of geodesies, and the role of the Christoffel symbols* 
appearing in (41) is thus far known to us only in connection 
with these world-lines. Einstein's reasoning is as follows: 

Let/ be a scalar or better a scalar field (i.e. an invariant 
function of position within the world). Differentiate it twice 
along any world-line. Then 

d 2 f df d 2 x t d 2 / dx a dxp 



ds 2 dx t ds 2 dx t dxp ds ds 

will again be an invariant. Let the line be a geodesic. Then 

x L j a/3 ( dx„ dxg , , . ... 

— = — < > — — , and the invariant will assume 

s 2 { <■ ) ds ds 



ds 2 
the form 



d 2 f .I d *f / a/3 I 3/ "I dx a dx, 



= r d2 f - \ a ^\^c\ 

L dx a dXa I L J dx L -J 



ds 2 I— dx a dxp \ L j dx L — 1 ds ds 

Since the contravariant tensor (of rank two) dx a dx$ is arbitrary 
(for from a given point a geodesic can be drawn in any direc- 
tion, i.e. with arbitrary ratios of dxi, dx 2 , etc.) and its product 
into the bracketed term is invariant, the latter, i.e. 

f =^L.-f«\ll (43) 

dx L dx K I a ) dx a 

is a covariant tensor of rank two. This proves the proposition 
for the special vector A L =df/dx t . To prove it for any 
covariant vector, notice that any such vector A t can be repre- 
sented by the sum of four (generally n) terms of the form 
\pdf/dx t , where \p and / are scalars. Thus it is enough to 
prove that 

dx„ \ dx t / \ a ) dx„ 



is a tensor. But this is equal to 



*Notice in passing that ^ > is not a tensor. 

X j 



(ovarian i Differentiation 61 

«. + ■*-*- 

OX, ()X K 

which, being the sum of covariant tensors of rank two 
itself a tensor of the same kind and rank. 

Thus the tensor character of the derivative (41) of any 
vector A, is proved. Notice that for constant g„ (galilean 
world) the Christoffel symbols vanish and this covariant 
tensor of derivatives reduces to an array of ordinary^dem a- 
tives dAJdx K . 

The proof of the tensor character of (42), which can be easily deduced 
from that of (41), may be left to the care of the reader. It is interesting to 
note that the covariant derivative of the metrical tensor g lK itself vanishes 
dentically. In fact, substituting in (42) g LK for B lK we have 



and since 
i< 






which will also be useful in other connections, and similarly for the last 
term, we have 

3g« 



["]-[*. X ] 



dxx 
But by the definition (13) of the symbols, and since g = g_ , we have 

Thus, g LK \ = 0, identically. 

Let A lK be as in (41), where A, stands for any covariant 
vector. Then, since the second term in (41) is symmetrical 
in i, k, dAJdx K — dA K /dx L = A lK —A Ki , being the difference of 
two tensors, is again a tensor. This tensor is called the rotation 
of the covariant vector A t , and can be written 

Rot (.4.) = i^L _ Mi. (45) 

dxp dx a 



62 Relativity and Gravitation 

This covariant tensor of rank two is manifestly antisym- 
metrical, i.e., in the case of a four- manifold, a six-vector. 
Notice that although the proof of the tensor character of the 
rotation was based on the metrical formula (41), yet the 
rotation itself, as defined by (45), is entirely independent of 
the metrical properties impressed upon the manifold. It 
contains no trace of the metrical tensor g LK . 

The same is true of a tensor of rank three which can be 
deduced from (42). Let in that formula B lK be an anti- 
symmetric tensor or six-vector. Then 

B^+B Al +B^- ***- + «3si + 23* . (46) 

d%x ox t dx K 

Thus the right hand member is again a tensor. This is called 
the antisymmetric expansion of the six-vector B tK . It will, 
together with the rotation (45), be of use in connection with 
electromagnetism. 

Another tensor derived from a six-vector of equal import- 
ance in the said connection is 

^ = VTg.£" (V-~g^) = DivG4' K ), (47) 

a contra variant vector, called the divergence of the contravariant 
six-vector A tK = —A" . The proof of its tensor character, to be 
based on (42), can be omitted here. 

Finally let us mention, without proof, that 

^7 ^7 (v:ig^)=div(^), (48) 

called the divergence of the contravariant vector A K , is a scalar or 
invariant. 

24. The Riemann-Christoffel tensor is of such capital im- 
portance for Einstein's gravitation theory, and for the 
geometry of any riemannian manifold, as to deserve to be 
treated at some length. 

It is a metrical tensor of rank four, built up of g tK and their 
first and second derivatives, known to the general geometers 
since the time of Riemann. 



The ( Curvature Tensob 

I i expresses the so-called curvature properties oi .1 mani- 
fold or n-space whose metrical relations arc fixed by th< 
&, , and to Einstein il served as the material for building up 

his gravitational field-equations. 

In order to arrive at this all-important tensor let US .>tart 
from an arbitrary covariant vector A t and let us write down 
its second covariant derivative, that is to say the covariant 
derivative of the tensor A lK which is the covariant derivative 
of A„ i.e., by (42), 

_ dA„ f i\\ ( k\\ , 

dx x ' a ' I a ' 

where A iK is as in (41). Similarly, transposing k and X, let 
us write the second covariant derivative 

I _ dA * _/ tK I a _ / x *l i 
dx K I. a ) { a ) 

Either being a third-rank tensor, so will be their difference 

A a _ dA lK dA lX ) t\ \ ,/**>•) 4 

d.\\ <1.\\ { a ) { a ) 

This is, by (41), 

LteA aT-J^Xa /J^ + Ll a ft »f~ 

In. the second term the indices a and /? over which the sum is 
to be taken can be interchanged. Thus A^ — A^ is the inner 
product, of an arbitrary covariant vector A a into the sum of 
the tw r o bracketed expressions. This sum, therefore, 

(49) 

». - & ( t\\ d j ck\ i c\\f ISk\ (ikH0X\ 

'"" dZ\ a / " ^\ a f + \ fi ! I « J " I (3 A - V 

is a mixed tensor of rank four. This is the Rieman n-Christoffel 
tensor which, for reasons to appear presently, may as well be 
called the curvature tensor. 



64 Relativity and Gravitation 

Strictly speaking, Riemann's own system of four-index 
symbols (iju> Xk), discovered in 1861 in connection with a 
problem in heat conduction (Mathematische Werke, 2nd ed., 
p. 391), is the purely covariant tensor associated with (49), 
to wit 

■^Xk = (tM. Xk) = g M a £?«x ■ ( 50 ) 

From this we have conversely, 

K-x = <T(tM, Xk). (50a) 

Also the latter tensor was used in geometry for a long time.* 
The Riemann symbols are, for an w-space, w 4 in number, 
and for our world, therefore, as many as 256. But they are 
bound to one another by the linear relations 

(ifJL, K\) = — (/JLL, KX), (ifl, KX) = — (l/JL, Xk), (t/Lt, «X) = (kX, i/x) , 
(l/JL, /cX) + (lX, /Ik) + ((.K, Xju)=0, 

so that the number of essentially different, i.e. linearly 
independent symbols is reduced to 

N= "W-V . (51) 

12 

For a proof see, for instance, Killing, loc. cit., p. 228. 

In the case of a one-dimensional manifold, a line, there 
is no such non-vanishing symbol. In fact, although a line may 
be ' curved ' from the standpoint of two- or more-dimensional 
beings in whose space it is imbedded, yet it has no intrinsic 
properties of its own to distinguish it from other lines, nor 
one of its parts from another. Take, for instance, a plane 
curve. If Aco be the angle between the tangents at two points 
separated by the arc As, the curvature of the line is defined 
as the limit du/ds. Now, this curvature is often called an 
intrinsic property of the line, because (unlike the sloping 
of the line) it is independent of a coordinate system laid in 

*Cf. for instance L. Bianchi, 1902, loc. cit., p. 72, where it is denoted by 
|ia, X/c|. The geometrical applications of the Riemann symbols are fully- 
treated in vol. I of Bianchi's work. See also W. Killing's Nicht-Euklidiscjie 
Raumformen, Leipzig (Teubner), 1885. 



Riemann Symbols 

that plane, yet it is entirely meaningless it the line is not 
conceived as a sub-domain of the plane. For bo is the angle - 
And from the bidimensional standpoint every curve is 

developable upon every other. 

In the case of a surface, n = 2, there is, by (51 1, essentially 
just one Riemann symbol, namely 

(12, 12), 

(21, 21) being equal, and (12, 21), (21, 12) oppositely equal 
to it, and all others being zero. This unique symbol divided 
by the discriminant g is a differential invariant of the surface 
(or of its metrical form ds' 2 = g n dxi ? +2gxndxidxo-\- g^dx* 1 ) . This 
invariant, 

K = < 12 ' 12 > = n2 - 12 > . 

g gll g22 - gl2" 

is the familiar gaussian curvature of the surface, its reciprocal 
being the product of the two principal radii of curvature. 
This is an intrinsic metrical property of the surface, requiring 
for its general definition or its numerical evaluation no refer- 
ence whatever to a third dimension. In fact, (52) contains 
only the metrical tensor components g LK and their first and 
second derivatives with respect to any gaussian coordinate 
system spread over the surface itself as a network of lines. 
The curvature thus defined, in general variable from point to 
point, can be evaluated at any spot by dividing the exc<.-~ 
of the angle sum (over a straight angle) of an infinitesimal 
triangle by the area of the triangle. Again, as is well known, 
the necessary and sufficient condition for a surface to be 
developable upon a euclidean plane or for its fundamental 
form to be holonomously transformable into 
ds 1 = dx- + dy- 

is the vanishing of K, i.e. of (12, 12)', and herewith the vanish- 
ing of the whole tensor of Riemann symbols. 

For a three-space there are, by (51), six, and for the world 
or space-time as many as twenty independent Riemann 
symbols. A five-space has fifty independent symbols, and so 
on. But, no matter what the number of dimensions, the 



66 Relativity and Gravitation 

Riemann symbols always represent the curvature relations 
of the manifold, and their vanishing" continues to form the con- 
dition of an important property of the metrical form of the 
manifold. 

To begin with the latter,. suppose all g u: are constant over 
a domain of the world. Then all (in, Xk), and therefore also 
all the components of the tensor jB". x vanish throughout the 
domain. This then is the necessary condition for a domain 
of the world to be galilean, i.e., for the line-element to be 
holonomously transformable into ds 2 = c 2 df — dx 2 — dy' 2 — dz 2 '. 
It was proved by Lipschitz that this, i.e. 

is also the sufficient condition for the said reducibility (to a 
form with constant coefficients). 

In the second place, concerning the curvature relations, 
consider a surface <r or a two-dimensional sub-domain of the 
world, or of any metrical manifold. More especially, let a 
be a geodesic surface. This can be defined as follows. From 
a point 0(x L ) draw two infinitesimal vectors d£ t , drj, and con- 
sider the pencil of infinitesimal vectors 

dx t = adi; L -{-(3dri L , 

where a, (3 are free coefficients. Draw from the geodesic 
lines with each of these vectors as initial direction. The 
surface thus constructed will be a geodesic stirface through 0. 
Its normal v will be defined by the infinitesimal vector dy L 
orthogonal to d^ and d-q l} i.e., such that 

guc dy L dL = gu; dy\d VK = 0, 

and therefore also g lK dy t dx K — 0. The geodesic surface <j = o v 
will thus be completely determined by 0, one of its points, 
and by its orientation, given by the normal just explained. 
The line-element of the manifold (world) will give for the 
line-element of this surface, at 0, an expression of the form 

ds 2 = /zn dp 1 2J c2hi» dpi dp 2 -{-Jh2 dpi 2 , 

and the gaussian curvature of a„ will be, as before, with 

h = h n h 2 i — /?i2 2 , 



RlEMANNIAN ( i i'\.\ l I RE 



67 



K = 



(12, 12) ; , 
h 



>2 



This is, according i<> Riemann's definition, the curvature <>] the 

manifold, at O, corresponding to the orientation v of the 
surface. The suffix //. has to remind us thai the Riemann 
symbol is to be formed with h m as the- fundamental tensor (of 
the sub-manifold a,). It remains to express (52a) in term- 
of the original tensor g u of the manifold and the vectors 
ei£,, dr] t defining the orientation of the surface. This gives 
ultimately* 



A' = 



1 



^' U\, Kn) t 



where 



,SlK 



fc = 2' 



.sX/i 



#. 


dij t 




d£ dy K 


d^ 


dr)\ 




d ^ dv,* 


#. 


drj t 




< dr, K 


dh 


dv\ 




#„ d V)l 






the dashed sums to be extended only over such combination- 
of the indices for which i<\ and at the same time k<h- 

This will suffice to show the role of the four-index symbols 
in determining the riemannian curvature of a metrical 
manifold of any number of dimensions. In general, the 
curvature will not only vary from point to point but will 
also be different for different surface orientations (v). In 
short, the manifold will in general be heterogeneous and 
anisotropic with regard to its curvature. Such, for instance, 
will be the world as the seat of a permanent gravitational held. 
On the other hand, a galilean domain, for which all (tX, ku) 
vanish, will have everywhere and for every orientation the 
curvature zero. In other words, it will be flat or homaJoidal . 
The next simple case is that of a manifold of positive or 
negative constant curvature, for which, that is, K r = K is 
constant and equal for all directions of v. It may be interest ing 
to note that the necessary and sufficient condition for the 
constancy and isotropy of curvature is 



*Cf. Bianchi, loc. cit., pp. 340-343. 



68 Relativity and Gravitation 

(iX, km) =K(g tK gKp—gw gxJ, (54) 

for all values of the indices i, k, X, fx. These are partial 
differential equations of the second order for the g llc with K 
as a constant coefficient. They exhibit the flat manifold, 
K = 0, as a sub-case. 

Other concepts connected with the system of riemannian 
curvatures K v , such as the mean curvature, will be given 
later on in connection with gravitational problems. Here our 
purpose was only to show that the curvature of the four- 
world and, in fact, of a metrical manifold of any number of 
dimensions, is a concept as definite and essentially as simple 
as that of an ordinary surface. The only difference is that of 
a possible anisotropy of K v for three- and more dimensional 
manifolds, whereas there is, of course, no such possibility in 
the case of a surface. 

We are now ready to explain the use made by Einstein of 
the Riemann-Christoffel or the curvature-tensor in constructing 
his gravitational field equations. These will occupy our 
attention in the next chapter. 



CHAPTER IV. 

The Gravitational Field-equations, and the Tensor 

of Matter. 



25. As was pointed out on several occasions, the funda- 
mental, metrical tensor g lK of the world determines, through 
the line-element ds 2 = g lK dx,dx K , its null-lines and its geodesies, 
and these, in virtue of the explained concrete representation, 
rule the propagation of light and the motion of a free particle, 
respectively. It remains to build up generally covariant laws 
or equations which would enable us to determine the metrical 
tensor g lK itself. Needless to say that in looking after such 
equations Einstein had in view, from the outset, the gravita- 
tional field. Of this he knew that to a certain degree of 
approximation it was represented by the (non-covariant) 
differential equation of Laplace-Poisson for the ordinary- 
potential, 

V 2 ft=-47rp, 

the gradient of the potential 12 giving the right hand member 
of Newton's (approximately valid) equations of motion. The 
equation of Laplace-Poisson being of the second order it was 
natural to look for a tensor containing the second deriva- 
tives of the metrical tensor components together with the 
g lK themselves and their first derivatives. 

Such a tensor lay ready in the treasury of the geometry 
of w-dimensional spaces since the time of Riemann, and it 
represented, moreover, certain intrinsic properties of any 
such metrical manifold, its curvature properties. This was 
the covariant tensor of the four-index symbols (tju. Xk) or the 
associated mixed Riemann-Christoffel tensor B^ x . It was 
natural, therefore, and Einstein himself relates to us that 
such was his first thought, to utilize for the purpose in hand 
this very tensor. 

69 



70 Relativity and Gravitation 

As we saw in the last chapter, the vanishing of this tensor 
expressed a simple and at the same time a profound feature 
of a metrical manifold, to wit, the ~ 2 n z (n 2 — 1) independent 
equations 

formed the sufficient and necessary condition for the flatness 
of the manifold. In our case twenty such equations form 
the necessary and sufficient condition for a world-domain to 
be essentially galilean, i.e., for its line-element to be holo- 
nomously transformable into c 2 dt 2 —dx 2 —dy 2 —dz 2 . 

A domain, therefore, in which there is no gravitational 
field, i.e., no premanent field of acceleration, will certainly 
be characterized by the generally covariant equations -B t " x = 0. 
The same equations might at first suggest themselves for the 
description of a gravitational field outside of matter. It will 
be seen, however, after a moment's reflection that they would 
be too stringent for such purposes. In fact, the field of 
acceleration surrounding the sun, say, can certainly not be 
transformed away holonomously. The said equations would 
thus be too stringent for such fields and, in fact, for any 
acceleration field which, in our nomenclature, is a permanent 
field, i.e., not to be got rid of by any holonomous transfor- 
mations of the coordinates. At the same time, the g LK to be 
determined are only ten in number, forming a symmetrical 
tensor of rank two, while the Riemann-Christoffel tensor is of 
rank four, and consists of twenty independent components. 
Such considerations led Einstein to require for the gravita- 
tional field outside of matter a set of broader equations, 
yet of the second order, and ten in number. For this purpose 
the symmetrical tensor derived from 2? t " x by contraction with 
respect to a, X naturally suggested itself. 

In fine, writing B" Ka = G LK , Einstein's field equations outside 
of matter are 

G lK = 0. (111°) 

That G lK , obtained by contraction from the mixed tensor 
of rank four, is itself a covariant tensor of rank two, we know 



Field Eoi \n" 



from the preceding chapter. Moreover, by (49), to be con- 
tracted wiili respecl to X, a, we have 

(56) 

or, alter some simple transformations which can here be 

omitted. 



c « - \ e M . ; _ i7. \ . / + 5 - • 

5 2 logV — gr j tK - "I dlog V — g 



o,, — 



dx t dx. 



(55a) 



now, | i ; 



so that S lK (which is not a tensor) 



/"la/' 

is symmetrical, and such being also the first two terms in 
(55a), G lK is seen to be a symmetrical covariant tensor, of 
rank two; G lK = G KL . 

Thus Einstein's field equations (III ), valid outside of 
matter, are ten in number, and such is exactly the number of 
the metrical tensor components g u . The field equations 
would then give us a system of ten differential equations of 
the second order for ten unknown functions g„ of the co- 
ordinates. As a matter of fact, however, there exist between 
the covariant derivatives G^ of the G lK and the derivatives 
dG/dx t of the invariant G = g K C7,,. four identical relations 
(based upon certain identical differential relations discovered 
by Rianchi), to wit 

G=^G^ = ^^-,l=L 2, 3, 4. (56) 

d.Y, 

Owing to these four identities, to which we shall have to 
return later on, only six of the field equations are mutually 
independent, leaving therefore four of the g^ or any four 
functions of the g w free or undetermined. Such, however, 
should from the general relativist ic standpoint be the ease. 



r - d 

OX, 



72 Relativity and Gravitation 

In fact, from this point of view one would expect beforehand 
the field equations or any differential laws to be such as to 
leave us a perfectly free choice of the system of coordiantes. 
Einstein himself, for instance, makes use of this freedom by 
putting in most of his formulae V — g=l, which, by (55a), 
reduces his field equations to 

-{:}+{';}{rH <*> 

and leaves him still a threefold freedom of choice. The latter 
can often be used with advantage by making gu = g2i = gzi = 0. 
It will be kept in mind, however, that the equations, such as 
(57), thus simplified do not retain their form under general 
transformations. They are only useful as tephnical devices 
offering some advanatges in the treatment of special problems. 
The generally covariant form of the field equations is only that 
obtained by equating to zero the complete or general value of 
G lK , such as (55) or (55a). 

26. In order to see the relation of Einstein's field equa- 
tions to the more familiar Laplace equation, let us evaluate 
the curvature tensor G llc for the case of a 'weak' field, i.e. 
differing but litle from a galilean domain.* 

Thus, using a quasi-cartesian system of coordinates, let 
the fundamental tensor differ but little from the galilean 
tensor g lK , i.e., as in (21), let 

Sue giK ~i I uc i 

where all the y tK are small fractions. Then the products of the 
Christoffel symbols in (55) will be small of the second order, 
and the tensor in question will be reduced to 



G LK =±( La \- ±\ LK ). 
dx K I a ) dx a I a ) 

)nd order terms, 

{"}-K , ;]-"2"['.'] 



Here, up to second order terms, 



*Notice in passing that all gravitational fields known from experience 
are 'weak' in this sense of the word. 



Field Equations 73 

and since g u = g n = g 3 x = — 1, g44=l, while all other "/ vanish, 

{7} ~ ["]•"■»■ 

{' 4 "}=ra- 

Thus, using the index i for 1, 2, 3 and summing every 
term in which i occurs twice over 1, 2, 3, we have the approxi- 
mate curvature tensor 



a 

dx 



;[7]-i["]+^<K]-K]>« 



In the present connection the only interesting component is 
that corresponding to i = k = 4. This is, by (58), and on sub- 
stituting the values (13) for the Christoffel symbols, 

where V 2 = is the well-known Laplacian + — -+- — . 

dXi 2 dxi 2 dx-r dx/ 

If the field is stationary, the second and the third terms 
vanish and Einstein's last field equation, (744 = 0, reduces to 
the familiar equation of Laplace 

V ? g 4 4 = 0. (59) 

At the same time, as we saw before (p. 36), the equations 
of motion assume, in absence of gu, the form of Newton's 
equations 

£*-.*!, (236) 

dr- dxi 

where the potential = - — go* differing only by a constant 

factor from gu, again satisfies Laplace's equation. The 
complete contents of Newton's law of gravitation, thus far 
outside of matter, appear as a first approximation to Einstein's 
field equations and his equations of motion of a free particle. 



74 Relativity and Gravitation 

27. The ten field equations G lK = are valid outside of 
'matter', i.e., as is expressly stated by Einstein, in such 
domains of space-time in which there is not only no matter 
in the ordinary sense of the word but also no electromagnetic 
field, or, in fact, no distribution of energy of any origin other 
than gravitational. Following Einstein's example the word 
'matter' will be used to cover all such cases. This will har- 
monise with the property of energy already familiar to us 
from special relativity,* namely of possessing inertia, an 
amount of energy U being equivalent to an inert mass U/c 2 , 
which, by the law of proportionality, is also its heavy or 
gravitational mass. 

As we saw before, the role of the newtonian gravitation 
potential 12 is, in a first approximation, taken over by the 
tensor component gu multiplied by — c 2 / 2 . The vanishing of 
Gu was approximately equivalent to Laplace's equation 
V 2 J2 = which holds outside of matter. Within matter 
Laplace's equation is replaced in the classical theory of 
gravitation by the more general equation of Laplace-Poisson, 

V 2 r>= -4ttp, 
where p is the density of mass in astronomical units. f Now, 
since Gu reduces approximately, in a stationary field, to 

— ^ 2 g iA =p — V 2 fi, the idea easily suggests itself to make 
c- 

G 44 == - -i^- , (60) 

c ? 

and to consider this as the equation or at least as one of the 
field equations within matter. But, needless to say, such a 
single equation would riot by itself serve any relativistic 
purpose. What is required is a system of ten equations, of 



*And partly even from pre-relativistic considerations, such as in 
Mosengeil's investigations on an enclosure filled with radiation or those 
made in connection with Poynting's light-pressure experiments. 

fit will be kept in mind that a mass in in astronomical units is denned 

m 2 . 

by = force, so that its dimensions are 

\m] = [length X (velocity) 2 ]. 



Tensor or Mattkk 



which this should be one. In oilier words, fix- tensor '/,. In- 
to be made equal or proportional to ;i symmetrical co- 
variant tensor of rank two somehow associated with 'matter' 
and having for its 44-component the density p or what ap- 
proximately reduces to the usual mass density and therefore, 
apart from a constant factor, to energy density. Now, such 
a tensor was familiar from the special relativity theory under 
the name of stress-energy tensor often abbreviated to energy 
tensor. The merit of having introduced this concept into 
modern physics is chiefly due to Minkowski and Laue, pre- 
ceded in non-ielativistic physics by Max Abraham. The 
energy tensor made its first appearance in electromagnetism, 
in connection with the ponderomotive properties of an electro- 
magnetic field,* as the symmetrical array or matrix 



5 



/ll /l2 /lS Pi 

fn'fnfu P2 

JSl fai fi3 Pi 

p\ pi p-i-u 








= 


/ 

p 


p 

— u 









consisting of the six components /,& =fki of the maxwellian 
electromagnetic stress, of twice the three components pi of 
electromagnetic momentum (or Poynting's energy flux) and 
of the density u of electromagnetic energy. The physical 
significance of this tensor or matrix was that its product into 
the operational matrix 

d d jL a 

dx dy dz dt 

P per unit volume and its 



lor 



gave the ponderomotive force 
activity Pv, 

-lor£= \P U P 2 ,P Z , Pv|. 

Later on its role was generalized for a stress, momentum and 
energy density of any origin, not necessarily electromagnetic, 

*Cf. Theory of Relativity, Macmillan, 1914, Chap. IX, especially p. 23S. 
In reproducing it here, with pi written for gj , I drop the imaginary unit 
and put c = 1. 



76 Relativity and Gravitation 

provided only that the force and its activity could be repre- 
sented in the form 

P= — V/— — , Pv = ■ — divp. 

dt dt 

From the special relativistic standpoint this array of ap- 
parently heterogeneous physical magnitudes was important 
as it transformed from one inertial system 5 to another S' as 
a whole, to wit by the operator A()A, where A is the funda- 
mental Lorentz transformation matrix of 4X4 elements and 
A the transposed of A. The developed form of the trans- 
formation equations of stress-momentum-energy need not 
detain us here.* 

The important thing in our present connection is that the 
said stress-momentum-energy array is a symmetrical tensor 
of rank two. And since such also is the contracted curvature 
tensor G^ , the idea naturally suggests itself to make G LK pro- 
portional to a symmetrical covariant tensor T lK , of which the 
first nine components 7n, T&, . . . T ss are of the nature of 
stress or equivalent to it, the components T i{ (*=i, 2, 3) replace 
the momentum, and the last component Ta is, or approxi- 
mately reduces to, an energy- or mass-density. But then it 
is by no means necessary (nor is it possible) to fix beforehand 
the exact physical meaning of the several components of such 
an energy tensor or tensor of matter (as it is often called by 
Einstein). Their significance has to be fixed a posteriori, 
through physical applications of the field equations aimed at. 

If T tK is a covariant tensor of rank two, then, as we already 
know, 

T=g LK r„ (6i) 

is a scalar, the invariant of T LK .f Such being the case, g lK T 
is again a symmetrical covariant tensor. Now, guided partly 
by guesses (originally at least) and partly by considerations 

*It will be found on p. 236 of my book quoted above. 
tSuch also was Laue's 'scalar' in relation to his Welttensor', i.e. the 
matrix .S. 



Field Eqi tnoNS 77 

of conservation of energy and oi momentum, Einstein wrote 
down as his general field-equations 

<L = - 5e. a\ K -\ gui d, in 

c- 

the factor — S71-, c 2 being so chosen as to give, in a firsl approxi- 
mation, the equation of Laplace- Poisson. 

In fact, as we shall Bee from the more definite form to be given pre- 
sently, in a first approximation, 

r.,., = 7'=p, 

so that (he last of (111) gives Cm p, as in (60). Einstein's own 

c 2 

coefficient differs from ours by the gravitation constant which i- here in- 
corporated into p, the density in astronomical units. 

The previous equations (111°), holding- outside of matter, 
are a special case of these general equations, for T lK = (), when 

also r=o. 

To be exact, Einstein speaks first of ' matter' as 'everything 
except the gravitation held' (he. cit., p. 802) and writes 
G lK =0 outside of matter in this sense of the word. But later 
on (p. 808), trying to justify the exact form (III) of his general 
equations, he states expressly that 'the energy of the gravita- 
tion field' (if there is such a thing) has also to 'act gravita- 
lionally as every energy of any other kind', in short that 
gravitation energy too has mass and weight. Thus, rigorously 
speaking, there is 'matter' everywhere, and the equations 
(111°) are valid nowhere, unless there is no gravitation field, 
when they are superfluous. In other words, gravitation itself 
contributes also to the tensor 1\ K . Its contribution, however, 
is practically evanescent, and this circumstance makes the 
equations (111°) physically applicable. 

But even the contribution to 7^ (», fc=i, 2, 3) of stresses 
within matter in the ordinary sense of the word (tensions or 
pressures) is practically negligible, and so is the contribution 
to 7^4 of the energy proper outside of molecules, atoms or 
electrons, and we may as well omit it in T i{ , and take, for a 
first approximation at least, 7'.i 4 = p, where p is the density 

*Principles to which we may return later on. 



78 Relativity and Gravitation 

of ordinary matter or approximately so, always in astro- 
nomical units. Thus the idea easily suggests itself to build 
up the tensor T lK for a theoretically continuous body (a fluid, 
liquid or solid) out of its local density and the velocity com- 
ponents of its motion. For although the gravitational effects 
of the motion of matter are exceedingly small, yet the mere 
desire of writing generally covariant equations, say, of hydro- 
dynamics,* prevents us from discarding velocities in this 
connection. Thus, neglecting stresses, etc., let us introduce, 
after Einstein, the scalar or invariant p as 'the density' of 

doc doc doco 

matter, and the four-vector of velocity — - . Then p — * — - 

ds ds ds 

will be a tensor of rank two, a contra variant one, however. 

Construct therefore, by the principles explained in Chapter III, 

the associated tensor 

?« = P &a &* — j 2 - , (62) 

ds ds 

which will be covariant and, manifestly, a symmetrical tensor. 
This is Einstein's energy-tensor or tensor of matter to be 
used in (III) whenever tensions, pressures, etc., are negligible. 
As a matter of fact, in view of the limitations of even the 
most accurate methods of observations now available, this 
particular tensor will cover, presumably for many years to 
come, all needs of the physicist and the astronomer with 
regard to gravitation. f 

The value of the invariant T or the scalar belonging to this 
tensor, which is defined by (61), follows at once. Since 
g lK g ia g K p =&g*ti =8 K ag K p=gaff , and g aP dx a dx p = ds 2 , we have, by 
(62), 

T = p, (62a) 

that is to say, the scalar of the tensor in question is the density 
of matter. 



*And this was undoubtedly one of the reasons by which Einstein was 
influenced. 

fThe case of hydrodynamics will be covered by subtracting from (62) 
the tensor pg iK , where p is the (invariant) hydrostatic pressure. 



1'iKi.i) Equations 79 

On the other hand, in a local rest-system, in which only 

(dxn/ds)' 2 survives and is equal to l/gu, we have 

*« - P » 

*'.<?., for instance, r 4 4 = p£44, and therefore, by (III), rigorously, 

G44 — T- gi\ P- 

In a first approximation (gif=rl) this gives (60) or the Laplace- 
Poisson equation, as announced before. 

The general field equations (III) may, independently of 
(02), be given a slightly different form. Multiply both sides, 
innerly, by g lK , and write 

G=fG„. (63) 

Then, since g lK g lK = 4, 

G = —T. (64) 

c- 

Substitute this value of T into (III). Then 

G lK -\ g%K G=- ifr., (Ilia) 

c- 

the required form of the field equations. 

Notice that C7, as defined by (63), is the invariant of the 
curvature tensor G iK . This invariant or rather one-sixth of it 
is called the mean curvature of the world, at the world-point in 
question. In fact, in the case of a three-space G would, apart 
from a mere numerical factor, be the arithmetical mean of 
the three principal riemannian curvatures, and this would 
still be the case for a manifold of any number of dimensions, 
at least if ds" be a definite, positive quadratic form. This 
justifies the name given to G above,* and equation (64), 
independent of the particular form (62) of T lK , teaches us that 

*For a certain special world to be treated later on G will be proved 
explicitly to be six times the smallest value of the (constant and isotropic"! 
curvature of three-space which it is possible to choose as a section of that 
four-world. (Cf. Appendix, A.) 



80 Relativity and Gravitation 

this mean curvature is proportional to the scalar of the tensor 
of matter and vanishes, therefore, outside of matter. More 
especially, if stresses, etc., be negligible, the tensor (62) 
comes to its right and we have, by (62a) and (64), 

G = — p. (62&) 

c- 

The mean curvature of the world is thus proportional to the 
density of matter. 

Notice that p/e 2 has the dimensions of a reciprocal area, 
and such also are the dimensions of G, and of all G LK , since 
the g lK are dimensionless, and the G LK are linear in the second 
derivatives of the g lK with respect to the coordinates, each of 
which is a length. The same remarks hold good with respect 
to the field equations (III). 

It may be interesting to notice, even at this stage, that 
the mean curvature in familiar matter, say, in water under 
normal conditions, is, comparatively speaking, not insignifi- 
cant. In fact, remembering that the gravitation constant 
is 6*658. 10~ 8 , in c.g.s. units, we have for water at normal 
density 

G= 8t 6-658.1(r 8 =r86.1(r 27 cm.- 2 
9.10 20 

What is technically called the world-curvature is one-sixth 
of this. Thus the radius of mean curvature defined by 
R= v 6/trwill be, for water, 

i? = 5-688.10 13 cm., 

i.e., about 570 million kilometers or only 3'8 astronomical 
length units. 

But it would be rash to conclude with Eddington* that a globe of water 
of about this radius 'and no larger, could exist'. In fact, what is known 
from geometry is only that the total length of every straight (closed) line 
in a three-space of constant and isotropic curvature 1/i? 2 , of the properly 
elliptic or polar kind, is ttR, so that the greatest distance possible in such 
a space is %ttR and the total volume of the space is tt-R 3 . But, unlike such 

*A S. Eddington, Space Time and Gravitation, Cambridge, 1920, p. 148. 



( i RVAT1 RE [NVAR] \m 81 

a space, the world lias a non-definite fundamental differential form, and 
its riemannian curvature depends upon the orientation of the geodesu 

surface clement. Thus a direct transfer of the properties of an ellipti- 
space upon the (watery) world is certainly illegitimate. Notice, mon o 
that G, and therefore R, is remarkable as a genuine invariant of the four- 
world and not of a three-space laid across it as one of an infiiiii y ol possible 
sections. The best plan for the present is, therefore, to see in it only such 
an invariant of space-time, within the world-tube of a mass of water. The 
few numbers were here presented only to give an idea of the ordi 
magnitude attainable by the curvature invariant in ordinary matter, con- 
sidered as continuous. To those who like to contemplate sensational result 9 
the best opportunity is perhaps afforded by the atomic nuclei. According 
to Rutherford the radius of the nucleus of the hydrogen atom is about one- 
two thousandth of that of the electron, i.e, U0- 16 cm., and its mass practi- 
cally equal to that of the whole atom. This would give for the density of 
mass a value 3.10 24 times the normal water density, and therefore a curva- 
ture radius within the nucleus V 3 . 10 12 smaller, i.e., R=82 cm. only! 
The moral would then be that nuclei of about this radius and no larger 
could exist, with the same density. Fortunately they are believed to be 
much smaller. 

But it is time to return to Einstein's equations ol the 
gravitational field in order to see some of their further pro- 
perties. 

28. Multiply the field equations (Ilia), identical with 
(III), by g"°. Then, denoting by G t a and T" the mixed tensors 
associated with G lK and T lK , i.e., writing 

G a = o Ka G etc 

and remembering that g*° £„ = g?=*8* , we shall have, with 
8tt 



K = 

c 2 



G?- : k8?G=-KT* 



If G" x be the covariant derivative of G", and similarly for the 
energy tensor, we have, contracting with respect to X = a, ami 
remembering the meaning of 5", 

t« _ i s « dG „ ^ dG n 
dx a d.\\ 

But, by (56), the right hand member vanishes identically. 



82 Relativity and Gravitation 

Thus, as a consequence of the field equations we have the 
four equations 

7^=0 (' = 1.2.3,4). ( 6 5) 

concerning the energy tensor or the tensor of matter. Thus 
also, out of the ten field equations only six are left for the 
determination of the potentials g lK , as announced before. 

The matter-equations, so to call (65), as a consequence 
of the field equations constitute a most remarkable result.* 
Notice that they are entirely independent of any special form 
of the energy tensor, such as (62). They are manifestly 
general, i.e., valid for any covariant tensor T lK , merely in 
virtue of putting it equal, or proportional, to the curvature 
tensor 

G lK — 2 g tK G, 

the left hand member of Einstein's equations. 

In order to see the significance of the equations (65) 
remember that T? a is the contracted covariant derivative of 
the tensor 

T a = o Ka T 

Thus, if pressures, etc., are neglected, when T lK has the value 
(62), we have, remembering that g Ka g ia g K p =£ g K p =g lft , 

rpa _ dxp dx a 

as as 

More generally, if we add to (62) a tensor p llc due to stresses 
and any agents other than molar motion of matter, we shall 
have 

t:=p:+ p ^P~ ^-, (66) 

ds as 

where p? = g Ka p lK is the associated supplementary tensor of 
matter, and 

*Of course, the left hand members of the field equations were chosen 
by Einstein so as to yield these four equations of matter, as can be seen 
by comparing his earlier paper, Berlin Academy, 1915, pp. 778, 799, with 
a later, improved paper, ibidem, p. 844. 



Matter- Equations 

the covariant conjugate of the contravariant vector dx a . 
Of the mixed tensor (66) we have to take the covariant 
derivative, as defined in (42a), and to contract it with resjx < I 
to o and the new index. Thus the equations (65) will be 



■L ta 



dx a 
But, as can easily be proved, 



W^+t a J 7 '" - 



{?}- 




(67) 



where g is the determinant of the metrical tensor. 

Thus, the four equations (65) become, in any system of 
coordinates, 



a 



Vp dx 



Wg Tt) - J l *\ T$ = 0, fr=i, 2, 3, 4) (65a) 



where the mixed tensor of matter T* is as in (66). The left 
hand member of (65a) is a four- vector 

■*■ t to 

which is called the divergence of the mixed tensor T?. The 
equations (65a) can thus be read technically: 

The divergence of the mixed tensor of matter vanishes. 

This, of course, does not enlighten us as to their signifi - 
cance. To see their physical meaning take any coordinate 
system for which g— — 1, so that 



1Z " \< 






(656) 



and consider the case of a weak gravitational field, for which 
the g lt differ but little from the galilean values, i.e., in quasi- 
cartesian coordinates, g n = — l+7n, etc., as in (21). In the 
expressions (66) for the energy tensor itself the y lK can be 
neglected altogether, so that 



84 Relativity axd Gravitation 

d% t = — dx L (»=i, 2, 3), d£ A = dxi , 
and 

Tl=p1-p-^ — a , (f=l,2,3) 

7- o . a i dx 4 dx a 

li =pl +p — — . 

as ds 

In the right hand member of (656) the 7 te cannot be disre- 
garded without anihilating that member altogether. For the 
Christoffel symbols vanish for constant g lK . But since their 
values are taken to be small of the first order, it is enough to 
retain in the right hand member only 

T i * = pf+ P (^y=Pf+ P , 

and since { l | ) == [ l | ] = f-^- , the four equations become, 

dx t 

if pi be negligible in presence of p, 

dx a ' dx, 

For small velocities, ds = dxi = cdt, and if ^-be the cartesian 
velocity components dxjdt, 

TJ=p^- - pVl \ Tf = p{--±p Vl Vo, etc.; Tf^pf- 9 —, etc., 
C 2 ' c 1 c 

and 

7V=^+^, Tt=tf+ PV °',etc; T£=p£+p. 
c c 

Thus, neglecting cpf and cpS in presence of the momentum 
(per unit volume) pV of molar motion of matter, the first three 
equations will be 

(pvr — c' 1 pi 1 )+ — (pviv 2 -c 2 pi 2 ) +... + — (pvi)=p — ,etc, 
dxi dx 2 dt dx 

where 12= — \c-g^ plays again the part of the newtonian 



Mai rER-EQC \ i [< 

jxitrnti.il. tml the fourth equation 

? (p*) + • »+ d( L=\ P d < 

/i.v, a.v 2 dx 3 dt dt 

or, in obvious three-dimensional vector notation. 

dp P 

d/ t- dt 

ami. with fi written for the three- vector c*^ 1 , pf. 

d i \ . d d , \ i i- * 

- (pwi)-f- (pfr)-f- ' (pfir.^ =(hvf ; — p — ,etc. 

c</ dxi dx-2 

The left hand member of the last written equation is equal to 

— (pr^ -: ■- {pz'i) +...+: -.-: .':': V 

dt dx x 

d v_i j- 

= {pVi)-rpv< . <:.: V 

dt 

or. it" or be the volume and 5w = p<5r the mass ot an individual 

, . . dfoiSm) „. .. , . 

element ot matter, equal to . Similarlv we nave 

dt . St 

dp . ,. , , dp , ,. d(8m) 

6/ dt dt . 8~ 

Ultimately, therefore, the approximate equations of matter 
are. with i. j, k as unit vectors along the coordinate a\<. - 

(y8tfi)-8r\idivf^} dwt t +kdivU]=im . Yi: A 

dt 



and 



d . N 99 8ni .... 

dt dt 



The first three equations embodied in the vector formula A 
are the equations of motion of a continuous medium* under 
internal stresses (tensions) fa =<Ppi, and under the action of 

*A deformable solid, liquid or Quid. 



86 Relativity and Gravitation 

the gravitational field of which the newtonian potential is 
again, as in the case of the approximate equations of motion 
of a particle, ft = — \c l g^. The fourth equation, (B), is, apart 
from the new term on the right hand, the familiar equation of 
continuity. 

In other words, the first three equations express the 
principle of momentum, — the amount of momentum acquired 
by matter from the field per unit time being given by 
8m . grad ft which is the newtonian force on the mass element 
8m. And the fourth equation expresses the principle of energy 
or, equivalently, of matter, — the amount of energy (c 2 8m) 
acquired by a material element per unit time being equal 
to the mass 8m of that element multiplied by the local time- 
variation of the potential, i.e., approximately equal to the 
decrease of the potential energy of that element. 

It is scarcely necessary to say that this gain or loss in energy or in mass 
of ' matter ' placed in a gravitational field, according to the sign of dft/oV, 
is immeasurably small. Its discussion in this place may have only a mere 
academic interest. If it be neglected, (B) gives at once the usual equation 
of continuity, and (A) assumes the perfectly familiar form 

pV— i divfi— . . . =p V ft = gravitational force per unit volume. 
On the other hand it is interesting to notice that the equation (B) becomes, 
for V = 0, at once integrable and gives 

8m=8m e~~ft/ c2 > 
where 8mo is 8m for ft = 0. A similar result followed from a gravitation 
theory proposed some time ago by Nordstrom (Phys. Zeits., 1912, p. 1126). 
Its interpretation may be left to the care of the reader. 

Returning once more to the rigorous equations (65a) we 
now see that the terms 

l a 



ff .Ti 

represent in general the momentum and energy (or mass) 
acquired by 'matter' in a gravitational field. 

The four equations themselves express the principles of momentum and 
of energy, as was made plain above on their appropriate form. I avoid 
purposely to call them principles of 'conservation' of momentum and of 
energy. For although Einstein succeeded in giving them the form* 

*Sitzungsberichte of Berlin Academy, vol. 42, 1916, p. 1115, where the 
German T and / are the above V — g T,"V —g t. 



RGY PWXCII 



d 
dx- 



[vr^(77+c)]-« 



in which they would deserve the name of conservation, yet the f* built up 
of the g* 1 ' and their first derivatives has not the character of a general 
tensor, but behaves so only with respect to a certain class of coordinate 
ins (for which g = — 1). In view of this it has not seemed necessary 
to quote here the values of the r* . Suffice it to say that since, unlike T* 
they contain only the gravitational potentials 'and their first derivatives), 
Einstein calls V — g t a ' the components of energy of the gravitational field ', 
and V —g T a those of matter, and reads the last set of equations: the total 
momentum and the total energy- of matter and of the field are conserved. 
The point under consideration is after all but a formal one, and we prefer 
therefore to content ourselves with the original equations (65ft), interpreting 
their second terms as momentum and energy gained (or lost) without 
attempting o locate them as such in the gravitational field before their 
passage to, or rather appearance in, matter. 

Historically, the position is this. In the special or restricted theory of 
relativity the principles of conservation of momentum and of energy- were 
expressed by the vanishing of the 'lor' or, in Laue's nomenclature, of the 
Divergence of a world tensor, this 'Divergence' being a four-vector whose 
components were transformed by the Lorentz transformation, the four 
equations themselves being thus invariant with respect to this kind of 
transformation. The tendency to imitate these principles of conservation 
in the generalized theory was but a most natural one. But the proper 
generalisation of that special Divergence in a theory- admitting any trans- 
formations of the coordinates is the Divergence denned in the general 
tensor calculus, i.e., the contracted covariant derivative 

of the mixed tensor of matter T*. This is a genuine four-vector, a covariant 
tensor of rank one, and the original generally covariant equations 

T a = 0, ' 

■a 

are the only appropriate expression of the principles of momentum and 
energy. Their expanded form is (65a] and this cannot in general be given 
the form of 'conservation laws'. Only for constant g^ , that is in a ga'.. 
domain, does it reduce to 

d 

r— T« = 
dx a • 

which is identical with the vanishing of what was called the Divergence of 

T* in the restricted relativity theory. All attempts to squeeze the brc 

Divergence T a into the narrower one seem artificial and useless. For 

conservation as an integral law, cf. Einstein, Berlin Sitsungsber., 191$. 

p. 44$. 



88 Relativity and Gravitation 

29. That the gravitational field equations (together with 
the equations of motion and those of the electromagnetic field) 
can be deduced from a single variational principle or, as it is 
called, a Hamiltonian Principle, was first shown by H'. A. 
Lorentz (Amsterdam Academy publication for 1915-16) and 
by D. Hilbert (Gottinger Nachrichten, 1915, No. 3), and later 
on by Einstein himself.* More recently Hilbert, Weyl and 
others have returned to this subject in a large number of 
publications, in some of which the importance of the Hamil- 
tonian principle seems to be unduly overestimated. 

Since this matter is, after all, of a purely formal nature, 
it will be enough to give here but a very brief account of 
Einstein's own treatment as developed in the paper just 
quoted. 

With dx as a short symbol for dx\ dx 2 dx s dx± Einstein 
writes the Hamiltonian principle 

V^g(G+M)dx = 0, (68) 

where G, M are invariants. Since V — gdx, the volume of a 
world-element, is invariant, so also is the whole integrand. 
Let M be a function of the g^ and of q ( and dqjdx a , where q t 
are some space-time functions describing 'matter', while G 
is assumed to be linear in d 2 g M "/dx a dx$ with coefficients de- 
pending only upon the g M „. Then, by partial integration. 



V g Gdx = 



Fda, 



Vg G*dx + 

where da is an element of the boundary of the world-domain 

dx (the particular value of the integrand F being irrelevant), 

and G* is a function of the g M " and their first derivatives only. 
Let it be required that the values of g"" and of their first 

derivatives should be fixed at the boundary. Then 5 Fd<r= 0, 

and we can write, instead of (68), 

*A Einstein, Hamiltonsches Prinzip und allgemeine Relativitatstheorie, 
Sitzungsberichte der Akad. der Weiss., Berlin, vol. XLII, 1916, p. 1111- 
1116. 



I Iamii.iom W PRIM III I. 

a \Vg (G*+M)dx~0, 

where the whole integrand depends upon the p/^and j, and 
i heir first derivatives only. Tims the variation ol the ■■" gives 

at once i he ten equal ions 



where 
and 



d dH* dH* dN_ 
dx a dp( a ) dp dp 

H* = V~g c*, X = V - 1 M 

p-r* Pw= 






dx a 

Now, let G \)v the curvature invariant, i.e., in our previous 

notation, 

Then, on performing the said partial integration, it will 
be found that 

With this value of H* the (•([nations (69) become identical 
with Einstein's field equations as given above, if we put 

f = —V—p T 

i.e., 









„„ dN 
d;< 


= — V 


— P " a 1 
.-. .Sep - 1 < 


or, 


in 


terms of 


the covariant tensor oi matter, 


\ 






dM _ 
dfff' 


-Tat- 





71) 

The verification of this statement may be left to the care of 
the reader who may confine himself to systems for which 



90 Relativity and Gravitation 

30. Gravitational waves. Let us close this chapter by 
briefly mentioning a method of approximate integration of 
the field-equations given by Einstein (Berlin Academy pro- 
ceedings for 1916, p. 688) which exhibits the propagation of 
gravitational disturbances. 

Let the g lK differ but little from the galilean values, in a 
cartesian system, say, or — in our previous notation— let 

where the y tK are small. Then Einstein's approximate solution 
of his^field-equations is 

7„=T , «-|5:7 / xx, (72a) 

where y\ K is the retarded potential of — 2k T lK , that is to say, the 
familiar particular solution 



r ^K 

I IK ~. 

4-7T J 



— T lK (x , y , z, ct — r) dx dy dz (72b) 

r 



of ' the wave-equation ' 

G-£-'*) y -— * r - (72c) 

In (72b) r is the three-dimensional distance of the point for 

which y\ K is required (for the instant /) from the integration 

element dxdydz at which the value of T lK prevailing at the 

r . 
instant t — — is to be taken. 
c 

This solution represents gravitation as being propagated 
with the normal light velocity c, the slight changes of the latter 
due to the gravitational field itself being manifestly neglected. 
In this approximation the rigorously non-linear field equations 
are replaced by linear differential equations of the form (72c), 
the usual wave-equations. 

In the sub-case of a stationary gravitational field, when 
the whole tensor of matter is reduced to T Ai = p, we have by 
(726), as the only surviving y' lK , 

, k pdxdydz _ kS2 

744 ~"2x J J J "~r " 27' 



Approximate Integration (, i 

where Q is the ordinary newtonian potential of the gravitating 

masses, and, by (72a), the only surviving y lK , 

Kit 212 

711 = 722 = 733= —744= — - — = — —i 
47T C 1 

so that, as before, the role of the potential ft is taken over in 
part by — %c 7 y u - 



—7 



CHAPTER V. 

Radially Symmetric Field. Perihelion Motion, Bending 
of Rays, and Spectrum Shift. 



31. In order to represent the motion around the sun of a 
planet as a 'free particle', of mass negligible compared to 
that of the central body, it is enough to find a radially sym- 
metrical solution of Einstein's field equations outside the sun, 

G lK = 0, (111°) 

considering the origin r = of polar coordinates r,<f>, 6 as a 
singular point. 

As a form of the line-element, sufficiently general for this 
purpose, let us assume 

ds%)= gi dr 2 ^ r-dtf - r 2 sin 2 </> dd 2 + g± cWt\ (73) 

where g\, gi, written instead of gu, gu, are functions of r alone, 
of which we shall thus far assume only that 

ki(») = -li «*(») = !, (74) 

i.e., that at distances r large compared with a certain length 
belonging to the sun (which will appear in the sequel) the 
line-element tends to its galilean form ds 2 = —dr 2 — r 2 (dcjr + 
sinW0 2 )+cW. 

Let us correlate the indices of the coordinates by putting 
x h x 2 , x z ,Xi = r, 4>, d, ct 
respectively. Then the metrical tensor in question will con- 
sist of the components 

gi = giO) , g 2 = - r 2 , g 3 =- r 2 sin 2 <£, g 4 = g*(r) , (73a) 

where g K has been written for g KK . 
In the more general case 

ds 2 = g K dx 2 , 

in which the g K = g KK are any functions of all the variables, we 
have, for the only surviving associated tensor components, 

92 



Radially Stmmei ri< Field 






„ kk = 



and i herefore, recalling the definition of the Christoffel symbols, 



k ) 



2g K Bx t 
2& dx. 



for all t, >. , 



for i=fc«c, 






while all other Christoffel symbols vanish. 

Applying these formulae to the more special tensor (73a), 
writing 

h=\og gi, lu = log gt 

and using dashes for derivatives with respect to r=Xi, we 
have the rigorous values of the only surviving Christoffel 
symbols, altogether nine in number, 

(11) J12\ = (U\ _ 1 (U\_ 

\ll-**'\2j \3/- r > 14 (- = ** 



|22\ r [23\ 



)33\ r . 2 , /33 V 

~\ -, ( — • - Slll"<P,-\ rt /■— oMIliip, -I 



= -§sin2«£; 



. J44\- 



N-i 



These values substituted into the general expressions (55) 
for the components G lK of the curvature tensor give zeros for 
all those having lt^k, while the remaining four diagonal com- 
ponents are, rigorously, 



r 

6^33 = sin 2 </> . GU 

G 44 = H feu + T (// 1 '+// 4 ')~1 . 

gt 1_ r J 



(77) 



94 Relativity and Gravitation 

Thus we have, according to the field equations for r>0, 
that is, outside of matter, for the two unknown functions g\, 
gi the three differential equations 

OZ, ' 

(a) A/'+iVfo'-Ai')- — =0 

r 

(b) £(J£'-Ai')+fi+l = 

(c) hi'+h4' = 0. 

The last of these equations gives /?i+/f 4 = log(gi g 4 ) =const., 
that is to say, by (74) , 

gigi = const. = — 1. 

Equation (b) now becomes gi-{-rgA=l or 

-f log[r(g 4 -l)] = 0, 
dr 

so that r(gi— 1) = const. = — 2L, say. 

Thus the rigorous, and the most general, solution of the 
field equations (b), (c) is ultimately, 

2£ /\ 2LN- 1 

, gi= -Ql- — y , (78) 

where L is any constant, some length, characterising the sun, 

i.e., here the singular point or centre of the gravitation field. 

As to the first field equation (a) it is satisfied identically by 

these values of gi, g 4 -* 

To express the constant L in terms of M, the sun's mass 

in astronomical units, we may apply the following reasoning: 

As we already know, in the approximate equations the 

c 2 
newtonian potential Sl = M/r is replaced by 744. Now, 

*In fact, since gigi= — 1, the left hand member of equation (a) becomes 

hi"+hi' 2 +2hi'/r, 
and this is, by (78), 

-^frTEF [u-*w-««l 

which vanishes identically for all r=t=2Z,. 



1=1' 

~ r 



Radially Syhmetrn Field 

in the present case, 744 = £'i — 1 = —2L/r. Thus .\f=< I.. 
whence 

l= M . n 

c* 

Ultimately therefore, the line-element (73) corresponding 
to a radially symmetrical field becomes, rigorously, 



ds i = (l- — \ftft"-M -—Xlr-r^dtf+sm^dd 2 ), 



(80) 



a form of the solution of the field equations first given by 
Schwarzschild (Berlin Academy proceedings, 1916, p. 189). 
As was already mentioned in Chapter IV, the dimensions of 
L as defined by (79) are those of a length. This length, which 
is sometimes called the gravitation radius of the given body, 
amounts for our sun to about 1*47 kilometers. Thus, for all 
applications of any actual interest, L/r is a small fraction and 
the coefficient of dr 2 can be replaced by — (1 + 2L/7). 

32. Perihelion motion. Let us now consider the motion of 
a free particle (planet) in the field determined by the line- 
element (80), that is to say, by the metrical tensor 

ft = - ( 1- —I, & = -r 2 , &= -r-sin-</>, g 4 = 



1 j 



The general equations of motion (15) with the values (76) of 
the Christoffel symbols become, for t = 2, 3, 4,* 



d°-4> _ 
ds* ~ 


2 
r 


dr 

ds 


d<t> 
ds 


d-d _ 

d?~ 


-K 


dr 
ds 


+c 


d 2 Xi 
~d? 


--W 


dr 
ds 





+ isin2*g) ; 



(fa J </s 



*Instead of the first equation of motion (1 = 1) it will be more convenient 
to take the identical equation g lK &, X K = 1. 



96 Relativity and Gravitation 

Lay the plane $ = 71-/2, the equatorial plane of the coordi- 
nate system, through the direction of motion of the planet at 
some instant / - Then, at that instant, d4>/ds = § and sin 2<f> 
= 0, and therefore, by the first equation, permanently = 71-/2, 
that is to say, the planet will describe a plane orbit, and the 
remaining two equations, together with the identical equation 
giK^iX K = l, will become 

6 + — 0=0, 

r 

x 4+/Z4' rxi = 0, 

g iXi *- -^-1*^=1, (80a) 

Si 

where /* 4 = log g 4 and g 4 =l — 2L/r. The first two of these 
equations can be written 

— log (r 2 e) = 0, — log (g 4 *<) = 0, 
as as , 

and give 

r 2 'd = p, giX4 = k, (81) 

where p and k are arbitrary constants.* With the values of 
±i and 6 derived from (81) equation (80a) becomes ' 



r 4 
or, putting p 



:© ! +(^)0-f) 



(±y. 



fe2_i or 

+ — p-p 2 +2L P 3 . (82) 

£2 £2 

The determination of the orbit is thus reduced to a quad- 
rature. As an alternative we may write down the differential 
equation of the orbit, by differentiating the last equation with 
respect to 0, 

*The first of (81) represents the slightly modified law of Kepler: areas 
swept out by the radius vector in equal proper times of the particle (s/c 
instead of t) are equal. 



Perihelion Moth 

P-+p=- L - +ZL,r. (82a, 

der- p- 

Hither equation differs from the familiar equations of celestial 
mechanics, based on Newton's principles, only by the under- 
lined last term of the right-hand member. 

It is well known that in the absence of this supplemental^- 
term the orbit is a conic (an ellipse, a parabola or a hypert 

p = — [l + ecos(0-«;] (83; 

P 1 

with fixed perihelion. w = const. In fact, equation (82j 
identically satisfied by (83) ; and so is (82) if we put 

^(l-e-j=L-(l-fe 2 ), (83a) 

so that the orbit is an ellipse, a parabola or a hyperbola 
according as k- is smaller, equal to or greater than 1. 

In general, for orbital velocities comparable with the light 
velocity, equation (82) gives d as an elliptic integral of p. to 
which corresponds a complicated non-closed orbit. Its dis- 
cussion may be left to the care of the reader.* Here it will be 
enough to consider small velocities such as occur among the 
planets of the solar system. The supplementary term is then 
small compared with the newtonian ones, and the problem 
can be solved approximately by a conic (83) with slowly 
moving perihelion. If dp dd is the derivative of p when u is 
kept fixed, and if the term with (do: dd)- is neglected, we have 



(dp V _ /dp_ dp dwX 2 _^_ /dp V i 
dd ) \dd ' du dd ) ~ \dd ) 



dp dp do: 
<dd / \dd da dd / ' \dd / dd do~: dd 



and since (dp dd)- itself accounts for the first three terms of 
the right hand member of (82), the perihelion motion will be 
determined bvt 



*Cf. A. R. Forsyth, Proc. Roy. Soc, XCVI1 1920 , p. Ho. also \V. B. 

Morton, Phil. Mag., XLII (1921), p. 511. 

tThis reasoning, aiming at the seculcr motion of the perihelion, pre- 
supposes the knowledge of absence of a secular variation of the eccentricity 
€. Cf. footnote on p. 99, infra. 



98 Relativity and Gravitation 

dp dp du _ 3 
89 du> dd 

Here (83) can be used with sufficient accuracy for p and its 
two derivatives, so that 

dco _ L 2 l+3ecos w+3e 2 cos 2 «+e 3 cos% 
dd ]&? sin 2 « 

where u = d — w. Integrating this from to 2-k over 6 or, what 
for our approximation is the same thing, over u, we shall have 
the secular perturbation 8u>, the motion of the perihelion per 
period of revolution. The second and the third terms of the 
integrand, having in the second and the third quadrants 
values opposite to those in the first and fourth, contribute 
nothing to the secular perihelion motion, and the same is 
true of the first term, since this is the derivative of the periodic 
function — cot u. We are thus left with 



K~ 3L2 

p* J 



cot 2 « du, 



and since — cot 2 w is the derivative of cot u-\-u, 

5u=?^l. (84) 

P' 2 
This being essentially positive, the secular motion of the 
perihelion is progressive, that is, in the sense of the revolution 
of the planet. 

If the orbit be an ellipse (e 2 <l) with semi-axes a, b, we 
have, by the original meaning of the constant p, 

2 dd _^ r 2 d9 _ 2irab 
ds ' c dt cT 

where T is the period of revolution, and by (83), 

~ "c 2 " ¥ ~ c 2 T 2 ' 

expressing Kepler's third law. Thus 



Perihelion Mono 

K 2ira~ 'lira 



p cTb cTVl-e* ' 

and (84) becomes 

24ttV , 

c 2 r 2 (l-e-) 

which is Einstein's formula for the secular motion of the 
perihelion of a planet, undisturbed by other planets, per period 
of revolution.* This formula gives for Mercury, per century, 
43" or 43" "1, coinciding most remarkably with the famou- 
excess of perihelion motion of that planet, unaccounted for 
by the perturbations due to the other members of the solar 
family of celestial bodies. Although the rival explanation 
based on perturbing zodiacal matter, due to Seeliger — New- 
comb (taken up more recently by Harold Jeffreys), cannot 
be considered as ultimately discarded, this is certainly a 
most conspicuous achievement, perhaps the greatest triumph 
of Einstein's theory, yielding the required excess without the 
aid of any new empirical constant in addition to the light 
velocity and the gravitation constant. As to the remaining 
planets, Einstein's formula gives for them secular perihelion 
motions too small to be either contradicted or confirmed by 
observation in the present state of the astronomer's know- 
ledge. In fact, the only other serious anomaly unaccounted 
for by newtonian celestial mechanics (unless Seeliger's theory 
is accepted) is the excessive motion of the nodes of Venus, but 
with this Einstein's theory is essentially powerless to deal, 
since it yields, for a radially symmetric centre of course, 
rigorously plane orbits. But even the outstanding node 
motion of Venus is generally felt to be much less important 
than Mercury's perihelion motion yielded so naturally by 
Einstein's theory of gravitation. 

*A more thorough analysis shows that this is the only secular pertur- 
bation, the eccentricity, the period and the remaining elements of planetary 
motion being unaffected by the deviation of Einstein's theory from that 
of classical celestial mechanics. Cf. W. de Sitter's paper in Monthly Notices 
of the Roy. Astr. Soc, London, 1916, pp. 699 ct scq., more especially sec- 
tion 17. 



100 Relativity and Gravitation 

33. Deflection of light rays. The propagation of 
light is given by the minimal lines ds = of the metrical 
manifold determined by the quadratic form (80). By reasons 
of symmetry it is again sufficient to consider the plane 
(j) = const. = 7r/2. Thus the light equation becomes 

— dr- + r W = g A c 2 df, g 4 =l-—. 

If v be the system-velocity or the 'coordinate velocity' of 
light, defined bv 

-(JMS)Mf)'. 

the preceding equation gives 

K(f) 2 -Kf)>-- 

and if t\ be the angle between the tangent to the light ray and 
the radius vector, so that dr / da = cos rj, rdd / da = sin -q, 

c- 1 [~COS 2 ?7 . „ ~| , , 

— = — + sin 2 rj . (85) 

v 2 gi L gi -1 

Thus, if the ray be radial, away from or towards the origin, 
the light velocity is cg4, and if transversal, c 'vgi, both principal 
velocities being smaller than c, and both tending to c at 
infinity. Neglecting the square and the higher powers of 
L/r, which even at the surface of the sun is a very small 
fraction, we can write, approximately, v/c = 1 — (l+cos 2 J7)L/r. 
The velocity of light being determined by (85), the shape 
of the ray or the light path between any two points 1, 2 can 
be found* by means of Fermat's principle 



dt=8 



^ =0. 



In fact this principle can be proved to hold, at least for 
stationary gravitational fields, i.e., for g lK not containing the 

*In terms of r, r\, and thence by integration in terms of r, d. 



Deflb noN 01 i< ■ 101 

time* as in the case; in hand. Those interested in such an 
application of Format's principle may consult de Sitter's 
paper quoted in the preceding section. 

But a much more speedy way of obtaining the light path 
is to consider it as the limiting case of the orbit of a fret- 
particle. In fact, returning to the differential equation (82a; 
of such an orbit, and remembering the original meaning of 
the integration constant p, 

_ 2 </0 
ds 
we have for light, or for a 'particle' which would everywhere 
keep pace with it, 

p= co, 
so that the differential equation of the light path becomes 

^ +p-3Lp 2 = 0. (86) 

dd- 

In the absence of the last term, which bears to p the small ratio 
SL/r, we should have p = po cos 6, a straight line whose shortest 
distance from the origin is r = 1/po, the angle being measured 
from the corresponding radius vector. Thus, replacing p in 
the last term by p cos 0, which amounts to neglecting IJp-- 
and higher order terms, we have for the light ray p = po [cos 0-+- 
Lpo(l +sin 2 0)J or 

— =cos0+ — (l+sin-0). 
r r 

The angle between the asymptotes (r/ro= °°) of this curve 
is easily found to be 

A = ^ = 4 ^. (87) 

r c«r 

This is then the total angle of deflection of a light ray arriving 

*For a simple proof see T. Levi-Civita's paper in Nnovo Cimento, vol. 
XVI, 1918, p. 105. Levi-Civita assumes also gn=gu = gn = 0. The latter 
limitation, however, does not seem to be necessary. Thus, for instance, 
it can be shown that Fermat's principle leads to a correct result in the 
case of a uniformly rotating system, i.e., obtained from a galilean system 
by the transformation d' = d-{-(j)t, co = const. 



102 Relativity and Gravitation 

from a distant source (star) to the earth, if r be, approxi- 
mately, the shortest distance of the ray from the origin, 
e.g., from the sun's centre. In the latter case, if R be the 
sun's radius, we have 4L/i? = 5'88/6"97.10 5 radians = 1"75, so 
that 

A=l"75- . 

This is Einstein's famous formula for the displacement of 
star images seen in comparative angular proximity to the 
sun's disc. It can be considered as fairly well verified by the 
results of the Eclipse Expedition at Sobral, Brazil,* of May 29, 
1919, which were ultimately estimated to give, when reduced 
to r = R, the value 1"'98 with a probable error of about six 
per cent. This is certainly more than a mere order-of-magni- 
tude coincidence, and speaks strongly in favour of Einstein's 
theory. 

The displacements according to Einstein's formula should, of course, 
be away from the sun and purely radial. The displacements measured 
on the Sobral plates deviated from radial directions, at least for four out 
of the seven stars, considerably, to wit by 35°, 16°, 8°, and 6° for the stars 
numbered 11, 6, 2, and 10, whose distances from the sun's centre were 
about 8, 4, 2, and 5R respectively. These deviations or the presence of 
transversal displacement components may well be due to the distortion 
of the coelostat-mirror by the sun's heat, as pointed out by Prof. H. N. 
Russell. Yet a refined investigation of this point during the next eclipse 
seems very desirable, and, as I understand, will be taken special care of 
at the Eclipse Expedition of September 20, 1922, at which it is designed 
to avoid the use of a mirror. The field of stars near the sun, during totality, 
will then be almost as favourable as in 1919-t 

34. Shift of spectrum lines. Consider an atom, say of 
nitrogen, placed in the photosphere of the sun, at rest or 
practically so. • Then its line-element or the element of its 
'proper time' will be, by (80), and writing for the present 5 
instead of s/c, 



ds 



— / 2L\ A 



*The measurements of the Principe Expedition, made under un- 
favourable weather conditions, seem by far less reliable. 

fSome preliminary details will be found in Monthly Notices of the Roy. 
Astr. Soc. for May 1920, p. 628. 



Spectrum Shim 103 

and any finite interval of its proper time 

Let another nitrogen atom he placed in one of our terrestrial 
laboratories, at a distance r from the sun's centre. Then its 
proper-time interval will be 

In particular, let A/i be the terrestrial, and At the solar time 
period of one of the natural vibrations or spectrum lines of 
nitrogen. 

Now, encouraged by the traditional belief in the somewhat 
vague 'sameness' of atoms of a given kind, Einstein assumes, 
as he did already in other circumstances in the special rela- 
tivity theory, that the said two atoms are 'equal' to each 
other in the sense of the word that the proper time?* of their 
vibration periods are equal to each other. Eddington in his 
Report (p. 56) simply says that an atom is "a natural clock 
which ought to give an invariant measure of an interval 8s, 
i.e., the interval 5s corresponding to one vibration of the atom 
is always the same". Weyl states the case in an apparently 
more profound way by saying that if the two atoms are 
"objectively equal to each other, the process by which they 
emit waves of a spectrum line, when measured by the proper 
time, must have in both the same frequency". 

In short, the founder of the theory, as well as his exponents 
assume, more or less implicitly, that 

As = Asi. 
If so, then the ratio of the solar to the terrestrial period of 
vibrations is 



£-0-.f )'(-!)■ 



or, since in our case R/r is but a small fraction, 

^L =1+L =1 + 2109.10-°. (88) 

Ah R 



*It is now usual to extend this name for ds/c from special to general 
relativity theory. 



104 Relativity and Gravitation 

Einstein's conclusion then is that the lines of the solar 
spectrum, compared with those of a terrestrial one, should be 
shifted towards the red, the proportionate increment of wave- 
length being 

*h = L =2-109.1(r 6 , 
X R 

or equivalent to a Doppler effect due to a (receding) source 
velocity of 0'633 kilometers per second. This amounts, for 
violet light, to about 0'008 A. Now, although with the 
modern means one-thousandth of an A or even less can be 
well detected in comparing spectra, Dr. St. John of the Mount 
Wilson Observatory, who observed 43 lines of nitrogen 
(cyanogen) at the sun's centre, and 35 at the limb, was unable 
to detect any trace of the predicted effect. His observations 
were made and discussed in 1917, and his final conclusion 
then was that "there is no evidence of a displacement, either at 
the centre or at the limb of the sun, of the order 0'008 A". 
Since that time, however, in view of the entanglement of the 
Einstein effect with shifts of a different origin, and seeing 
that the results of other astrophysicists were not quite so 
definite, Dr. St. John suspended his final judgment and is 
now taking up a thorough discussion of the whole material 
of solar spectrum shifts from E. L. Jewell's first observations, 
made about 1890, up to the present. The natural impression 
now is that it would be premature to either assert or deny the 
existence of the gravitational spectrum shift. 

Einstein himself has, on more than one occasion, expressed 
the very radical opinion that, should the shift be absent, the 
whole theory should be abandoned. Yet, in view of the hypo- 
thetical nature of the sameness of atoms in the explained 
sense of the word, such an attitude, though personally in- 
telligible, is by no means necessary. It is true that the in- 
variability of an atomic ^-period of vibration in a gravitational 
field can, with the aid of the equivalence hypothesis, be re- 
duced to its invariability while the atom is being moved 
about, — a property of atoms as 'natural clocks' already 



Na'm ral Cloi b 106 

utilised in special relativity.* Yd we do not know whether 
the atoms actually possess even the latter property. Thus, 

Einstein's intransigent attitude proves only the Btrengtl 

his belief that the atoms are or will turn out to he such 
natural, ideal clocks. But, after all, this is only a gu 
A very reasonable one to be sure; for if not among the atom-. 
then there is indeed but little hope to find such clocks among 
other 'mechanisms', natural or artificial. 

At any rate, a final astrophysical verification of Einstein'- 
spectrum-shift formula, supported perhaps by repeated 
experiments on canal rays, would be an achievement of 
fundamental importance. Until then 'the natural clock' will 
remain a purely abstract concept. 

*It is this theoretical attribute of atoms which has led to the conclusion 
that moving hydrogen atoms (canal rays) will emit, in transversal direc- 
tions, waves (1— v 2 /c 2 )~' / * times longer than atoms at rest. But even this 
shift effect, though tried experimentally, does not seem to have ever been 
detected. 



CHAPTER VI 
Electromagnetic Equations 



35. Maxwell's equations of the electromagnetic field in 
empty space supplemented by the convection current pV, or 
the fundamental equations of the electron theory are, in 
three-dimensional vector notation, with x i = ct, 

— + curlE = 0, divM = 
dXi 

r- curl M = p — , div E = p. 

dx± C 

They contain, apart from the velocity v of moving charges, 
but two vectors E, M which may be provisionally called the 
electric and the magnetic forces. As is well-known from the 
special relativity theory, these equations retain their form or 
are covariant with respect to the Lorentz transformation, 
i.e., in passing from one to another inertial system.* 

They are not, however, generally covariant, and thus not 
appropriate to the purposes of the general relativity theory. 

What is covariant with respect to any coordinate trans- 
formations is the somewhat broader system of equations, 
containing two more vectors D and B which may be called 
the electric and the magnetic polarizations, f 



aB 

dXi 



+curlE = 0, divB = 0, (A) 



- — +curlM = p-, divD = p. (B) 

dXi c 

*Cf. for instance my Theory of Relativity, 1914, Chap. VIII, and, for the 
historical aspect of the subject, Chap. III. 

fOr the electric displacement and the magnetic induction respectively. 

106 



Electromagnets Eqi atio iot 

In a galilean domain or an incriial system D and B reduce to 
E and M respectively, bul in general, in a gravitational field 

or a non-incrtial system, the polarization> differ from the 
forces, being some linear vector functions of the latter 

The general covariance of these two groups of ell 
magnetic equations was first noticed and developed by 
F. Kottler as early as in 1912* and shortly afterwards, with 
due acknowledgement, incorporated by Einstein into the 
physical part of his general theory of relativity. 

Let F lK be an antisymmetric covariant tensor of rank two 
or a six-vector, which will embody in itself B and E, and thus 
may be called the magneto-electric six-vector. Then the group 
(A) of equations can be replaced by the equations 

dx x dx t dx K 

which are generally covariant since their left hand members 
are, by (46), Chap. Ill, the components of a general tensor of 
rank three, the antisymmetric expansion of the six-vector F lK . 
To compare (Ai) with (A) and to see the simplest form of 
the correlation between B, E and the six components of 
F lK use cartesian coordinates or, in the presence of a gravita- 
tional field (always 'weak'), quasi-cartesian coordinates and 
denote by 1, 2, 3 the rectangular components of B, E along 
the three axes. Then the group (A) of equations will be 

dB, . dE, BE* n , 
— + — — — =0, etc. 
dXi d.Yo dx 3 

dxi dx-: dx 3 

where 'etc.' means two more equations by cyclic permuta- 
tion of the suffixes 1, 2, 3 only. On the other hand, writing 
out (Ai) and remembering that F iK = —F a . we have 



*Friedrich Kottler, Raumzeitlinien der Minkowski? schen Welt, Sitzungs- 
berichte Akad. Wien, vol. 121, section Ho, pp. 1659-1759. 



108 Relativity and Gravitation 

dXi dx 2 dx s 

dF 23 dFzi dFn =Q 
dXi dx 2 dx 3 

and these four equations become identical with those just 
written if we put 

F 23 , Fzi, F n = Bi, B 2 , B s 

Fu, F 2i , Fzi — Ei, E 2 , E 3 

respectively, or more compactly, if i, k be reserved for 1, 2, 3 
only, 

F ik =B; F H = E. (89a) 

This then is the required correlation for the case in hand. 
Non-cartesian coordinates will be dealt with in the sequel. 

Next, let F lK be the supplement of F aP defined, as in (34), by 

F-^C^F^. (90) 

Then the group (B) of the electromagnetic equations will be 
replaced by the four equations 

Vg dX K 

where C is a contravariant four-vector. Such also being the 
left hand member, the divergence of F lK , as in (47), the equa- 
tions (Bi) will be generally contravariant. To compare them 
with (B) and to find the correlation proceed as before. Thus, 
on the one hand, 

_ dDi dM 3 _ dM 2 _ Vi 
dx± dx 2 dx 3 c 

Mh &D2 dD 3 _ 

dXi dx 2 dx 3 

and on the other hand, remembering that F KK = and F lK = 
-F K \ 



Electromagnetic Equations 109 

if —g C\ etc., 

d . . 

— (V-g F 41 ) + etc. = V -g C*. 

OX\ 

The required correlation is, therefore, 

V^(F<>, F<\ F«)=D U D„ A 

vCj (/?* F 3 \ F l -) = M 1} M 2 , M 3 

or, in the previous abbreviated notation, 

V^7f 4 * = D; V^F ik = M. (896) 

Since F" is thus seen to embody the electric polarization and 
the magnetic force, it may be distinguished from its supple- 
ment by the name of the electro-magnetic six-vector. At the 
same time we have, by comparing the right-hand members of 
the two forms of equations, 

V~ g {c\c\c>- c*)= P (jj, v f, * ; i) 

or, more shortly, 

C; C*= -4=,( V -- 1 Y 



(91) 



exhibiting C* as the electric four-current. It is interesting to 
note that since we can put v i /c = dx i ,'dx i and dx A dx A =\. 
the last correlation can also be written 

C*=-4= ^f. (91') 

V — g d.\u 

Since dx K is a contravariant vector as well as the four-current, the. 
factor of dx * will be an invariant, and since V — g dxj dx°dx s dx 4 is also 
an invariant, the volume of a world-element, we see that the electric charge 
8e=pdxidx>dx$ is again an invariant. Then, however, not p itself but p 
divided by the determinant — |gi*| will be the system-density of electricity. 



110 Relativity and Gravitation 

It may be well to illustrate the general transformation formulae of 

Fin r 

p, _dx„ dxp F 

by writing them out for the simplest case of two inertial systems S, S' in 
uniform translational motion relatively to each other. The transformation 
is in this case the familiar Lorentz transformation, i.e., in cartesian co- 
ordinates and with the X\ axis along the direction of motion, 

xi = y(xi'-\-(3xi), xt = Xi', xz=x% ', xt = y(Xi-\-fixi'), 

where fi=v/cand y = (1— j8 2 ) ~ **, if v be the velocity of S' relatively to S. 
First of all, since in this case the g LK have their galilean values (in both 
systems), we have 

B=M, D=E, 

so that there is no need to consider the supplement of F lK ; it is enough to 
treat F lK itself. Next, since X2, x s depend only on x 2 , x 3 \ being equal to 
them respectively, we have 

dx a dxa 
F' 23 = M 1 '= r-* t-2, F afi = F 3t = Mi. 

0x2 0x3 

ox 

F'. 3 i=~ F Sa =y(F 3l +l3F Si ), 

OXi 



Similarly, 



M 2 ' = y((3E 3 +M 2 ), 

and so on. Thus we get the transformation formulae 

Mx'^Mi, M-/=7(M 2 +/3E 3 ), M 5 ' = y{M z -^E 2 ) 

E^Eu E 2 '=y(E 2 -PM s ), E 3 ' = y(E 3 +^M 2 ), 

familiar from the special relativity theory. The corresponding transform- 
ation of the four-current may be left as an exercise for the reader. 

It will be kept in mind that the correlations of the forces, 
the polarizations and the current and charge density to the 
two conjugated six-vectors and the four-current given in 
(89a), (896), (91) are valid only for the particular case of a 
cartesian or quasi-cartesian coordinate system. With other 
systems, such for instance as the polar coordinates, even in a 
galilean domain, the correlation formulae are more compli- 
cated, and contain besides the determinant g the several 
components g lK of the metrical tensor or (in a non-galilean 
domain) parts of them, as will be seen later on. It is important 



Electromagnetn Equations ill 

to understand that there is nothing general about th 
correlations, apart from the faet that I\ K embodies somehow 

(he three-veetors B and E, and F u the vector- D and M, and 
C K the convection current and the charge density, everything 
being entangled with the metrical tensor and through it also 
with gravitation. 

From the standpoint of general relativity the n 
equations are henceforth no more the broadened maxwellian 
equations (A), (B) but the set of generally covariant or 
contravariant equations (Ai), (BO with the metrical link (90; 
between the two six-vectors. It will be well to gather here 
these somewhat scattered equations; the whole generally 
covariant electromagnetic set is thus 

dF lK dF± f dF Xt _ Q 



dx x dx t dx K 

1 d I / x 

V? OX K 



F' K = o La o KP F 

1 t> & •* a, 



(IV) 



This will read as follows: the expansion of the magneto- 
electric six-vector vanishes; the divergence of the electro- 
magnetic six-vector, the supplement of the former, is equal to 
the electrical four-current. 

36. The four-potential. Manifestly, the first of the equa- 
tions (IV) will be identically satisfied if we put 

F lK - d -± - *+±, (92) 

dx K d\\ 

where </> t is a covariant vector. If this be substituted, the 
six terms destroy themselves in pairs, and the covariant 
nature of (j> t ensures the required tensor character of F iK , the 
rotation of 4>, (cf. p. 61). The latter, which is seen to embody 
Maxwell's vector potential and the electrostatic potential, is 
called the four-potential. 

With the correlation (89a) the six equations (92) become 



112 Relativity and Gravitation 



b 



d /d d d \ 

-CUrl (01, 02, 3 ), E=— (01, 02, 0a) -( t— , x—, t— 104 

co7 \axi 0*2 ox 3 / 



dA 
B = curlA, E=- — -V0, 
coV 

exhibiting the three-dimensional vector A = — (0i, 02, 03) as Maxwell's 
vector-potential and = 04 as the electrostatic potential. 

The first group of equations (IV) being thus satisfied by 
(92), the second group gives 

;HK^'(S -£')> c ' (93) 

which, assuming g iK to be known, are four differential equa- 
tions of the second order for as many components of the four- 
current. Since the four-potential enters only through its 
rotation, we can without loss to generality subject its com- 
ponents to a kind of solenoidal condition, as follows. If 
4> K = g Ka a be the associated four-potential, a contravariant 
vector, then its divergence defined by (48) is a general in- 
variant or scalar, and the condition in question can be written 

£- K (Vg>)=0. (94) 

In a galilean domain the equations (93), (94) become 

d 2 A „ 9A 1 d 2 

— -v 2 A= — pv, — - — -V 2 = p 
c 2 dt 2 c c 2 dt 2 

divA+-i , il = o J 

c dt 

the familiar equations of the electron theory for the vector 
potential A= — (0 1; 2 , 3 ) and the electrostatic potential 
= 04. In general, however, the equations (93) for the four- 
potential will contain in a complicated way the components 
of the metrical tensor, which again means an entanglement 
of the electromagnetic with the gravitational field. This 
mutual relation of the two fields appears directly in the third 
of equations (IV) giving the general connection between the 
magneto-electric six-vector and its supplement. 



The Four-Potential 113 

Since the four-potential is a covariunt and dx t a COntra- 
variant vector, their inner product 

dl = 4>«dx K 

is an invariant. This invariant linear differential form plays 
the same role with respect to electromagnetism as the quad- 
ratic differential form 

d^-g ut dx t dx K 

with respect to gravitation. As the latter determines, inter 
alias, the gravitational field, so does the former determine the 
electromagnetic field. This is only a different way of stating 
that the <j> K , the coefficients of dl, determine the electromag- 
netic, similarly as the g lK determine the gravitational field 
together with the riemannian metrical properties of space- 
time. Recently a differential geometry somewhat broader 
than Riemann's was proposed by Weyl who goes deep into 
the matter and attributes to the linear differential form an 
equally fundamental metrical (gauging) function as to the 
quadratic differential form. But reasons of space prevent us 
from entering here into this subject, and the interested reader 
must be referred to Weyl's own book* for further information. 
Moreover, these new physico-geometrical speculations, 
although undoubtedly attractive, are still being debated 
between Weyl and Einstein, f and may therefore be appro- 
priately omitted in a book of the present type. 

37. Let us once more return to the electromagnetic equa- 
tions (A), (B) in order to compare them with the tensor 
equations (IV) for the case of a non-cartesian system of space 
coordinates. As a good example of this kind we may take 
any orthogonal curvilinear coordinates Xi, .v_>, x%. It is well 
known that if the space line-element in these coordinates be 
given by 

«fo»-f*g! + ^L + ** =^L (96) 

Wi 2 Tc'o'-' 10a 8 icr 



*H. Weyl, Raum-Zeit-Materie, 3rd ed., Berlin 1920, §10 anil §34. 
fCf. Einstein's remarks to Weyl's paper, with Weyl's reply, in Berlin 
Sitzungsber., 1918, and Einstein's recent paper, ibidem, 1921, pp. 201-204. 



114 Relativity and Gravitation 

and if R. : be the components of a three-vector R tangential 
to the ^',-lines of the network. 

Ld.Vi \W2W3S 8x2 \u' 3 U'i' dx 3 VaPtfOgX—l 

and the curvilinear components of curl R are 

(curl R^w.wTAf^ _ l/*Yl etc. (98) 

With these expressions the group (A) of equations becomes, 
provided of course that the w\ are independent of time. 



±(*-) + ±(h\-±.(^). . exc .. 

dXi \u- 2 w ?J / dx s \ Wz ' dx 3 \U' 2 ' 

L(Jl\ + ±(*l) + A. fA) = „, 

dXi \Wo "d}?y bXn \IL'3 W\* dXz \u-'l U'o ' 

and similarly for the group (B) of electromagnetic equations. 
These equations are to be compared with the first and second 
of the tensor equations (IV). To find the required correlation 
in terms of the g^ notice that if the domain is assumed to be a 
galilean one,* we have 

ds 2 = g LK dx t dx K = dx± 2 — da 2 , 
so that 

1 - 

git =— — , £44=1, 

and the remaining g lK vanish. Under these circumstances 
the comparison gives at once, with g t written for g ih 



*Otherwise, say in the presence of gravitation, not the whole of — is 

m 

to be thrown upon the coefficient of dxi in the expression for the length 
ddi considered from the system-point of view. 



THOGON'AL COOR. 



11 



1 1 

F*= -7= -If i. etc.; f = 






o = 



(K\ 



\ 






. etc.. = p. 



■:. 



which is the required correlation. 

The relations between the polarizations and the forces. 
determined in general by the third of equations (IV), follow 
ict, since in the present case g" = =1. and 

the remaining g^ vanish, we fa 



F* 



: 



that is to s 



j 






1 = - F m 



and therefore, by = 

B=M. D = E. 

the polarizations are identical with the : :ua- 

tior.- A B reduce the usual electromagne: :ons 

for the vacuum, giving - j> it veloci: - 

result might have been expected 

from that correspor. ling - - 

by the use of curvilinear ins: ; - - 

38. Let us now consider the relation be B. D and 

M. E in another example which, bes es g " 

in a general way. will show bow 
mag: - as is 

If a system be.- .=..= . 

dimensional line-element can be 



is 8 = ■ 



- 



In a weak g . . - 

but litt!. 2 ? " . - 



. 



116 Relativity and Gravitation 

cartesian or quasi-cartesian coordinates, gu and the ga will 
differ but little from + 1 and — 1 respectively, and the remain- 
ing g ik will be small fractions. Thus, from the system-point 
of view,* the electromagnetic equations (A), (B) will be , 

«2i + ™1 _ ^=0,etc, 

d%i dx 2 dX 3 

so that a comparison with the tensor equations will give again 

F ik =B ) F ii = E;F ik =- 7 =m, /*= "7=D , (89) 

v-g V^-g 

as in (89a), (896). 

Since g 4 ; = 0, the general relation F lK = g ia g K p F afi between 
the two six-vectors will now give 

F 23 = F 23 (g 2 2 g33 ~ gj + ^ 31 (g 2 3 go. - &1 g3 3 ) + ^ (g21 gS2 ~ gll g3l) 

and two similar equations for F 3h F i2 . But these are the solu- 
tions of the three equations 

F 2S = — (g n F 2S -\-gi 2 F 31 -\-gi 3 F 12 ), etc., 
h 

where h is the determinant | g ik | . Now, h = g/gn, and therefore 

F 2S = ^ (gui^+^i+gw/^), etc. 

Again, 

Fu = g4 a g v F* = g ii gu F*\ etc, 
i.e., 

^4i = gu(gnF* +gi 2 F 42 +g 13J F 43 ) 

and two similar equations for F i2 , F 43 . Whence, by (89), 

gu 

Mi=- ~ (gnBi+gi 2 B 2 -\-g 13 B 3 ), etc. 

V-g 

Et= p^ignDx+guDz+gM, etc, 

V-g 



^Analogously to the sense in which 'the system- velocity' of light was 
used previously, and contrasted with the local point of view. 



Light in Gravitation Field 



117 



or, solving for the polarisation components and noticing that 
l/g«= A' 44 , 

Dy= -V^g. g u (g u Ei+g*E a +g»E a ) l etc. 

Thus B is exactly the same linear vector function of M 
as is D of E. Introducing the symmetrical linear vector 
operator 



(101) 





£ u 


gM gM 


— u> = 


2l 

S 


a- 2 a 23 




gtl 


„32 ? 33 


we can write shortly 






B = M M, 


d=a:e, 


where 







tl = K = V-g. g «. 



(102) 
(103) 



In absence of gravitation the g M assume their galilean values, 
the operator d> becomes an idemfactor, g 44 =l, and p=K = l, 
giving B = M and D = E. From the system-point of view the 
vacuum is thus transformed by gravitation into a crystalline 
electromagnetic medium with anisotropic permeability m a "d 
permittivity K. These operators have, however, by (103). 
at every point common principal axes (which are orthogonal) 
and the same principal values. Now, owing to this peculiars ty 
the velocity of propagation of an electromagnetic wave, 
although varying from point to point and dependent upon 
the direction of the wave-normal, can be easily proved to be 
independent of the orientation of the light vector D. Thus. 
although the medium is anisotropic, there will be no double 
refraction due to the gravitational field.* In fact, if n be the 
wave-normal and t> the velocity, that is the system-velocity 



*Cf. in this connection A. O. Rankine and L. Silberstein, Propagatioti 
of light in a gravitational field, Phil. Mag., vol. 39, 1920, p. 586. 



118 Relativity and Gravitation 

of propagation, along the wave-normal, we have from the 
electromagnetic equations (^4), (B), (102), wilh p = 0, 

— iOS = VMn, - l xM= FnE, (104) 

as will be seen at once by considering a wave of discontinuity 
and using the general compatibility conditions given else- 
where.* Now, since the operator K is identical with y, the 
last two equations give 

- KE+Vn(K- 1 VriE) = 0, 

for every direction of E. Here the operator K ' 1 is the inverse 
of K. If Ki, etc., be the principal values of K and «i, etc., the 
components of n or the direction cosines of the wave-normal 
with respect to the principal axes, the last equation gives at 
once 

&_ nKn = K ini *+K 2 n 2 2 +K 3 n 3 2 

i.e., a propagation velocity independent of the orientation of 
the light vector, which proves the statement. 

If gi, g 2 , g s are the principal values of the vector operator 
gn, gn, • • • g33, the inverse of -co, then the principal values of 
-co itself are 1/gi, etc., and we have, by (103) and since 

g = glg2g3gU, 

(105) 



c 2 - 



Cwr . n 2 - . n£ \ 
— + — + — • 
gl g2 g3 - 1 



Such being the formula for the velocity of propagation on 
the electromagnetic theory of light, it is interesting to com- 
pare it with the light velocity v yielded directly by Einstein's 
fundamental equation ds = 0. This velocity is taken along 
'the ray' instead of the wave-normal. Thus, by (100), if u 
be a unit vector along the ray, and u t its direction cosines, 

*L. Silberstein, Annalen der Physik, vol. 26, 1908, p. 751 and vol. 29, 
1909, p. 523, or Theory of Relativity, London, 1914, p. 56. 



PONDEROMOTTVE FOW E 119 

< r/x; dx k 

— = - gik r- — = -iik*iVk, 

v da da 

and especially it //, be the direction cosines with respect to the 

principal axes of the operator g n , ga, . . . gu, 

c 2 1 

— = ki«i 2 +gs«2 2 -fg»«g*J- (106 ) 

v 2 gu 

Formula (85), used in connection with the bending of rays around the 
sun, is only a special case of (106). In that case the principal axes are 
along the radial and all the transversal directions, while the principal values 
are 

1 1 

gl=— - =— ~ , g2=g3=—l, 

g*i g* 
and u l 2 = cos 2 rj, w. 2 2 +W3 2 = sin 2 ?7, so that (106) reduces to (85). 

If the wave-normal n coincides with a principal axis, say 
with the first one, we have, by (105), t) 2/ c 2 = — g«/,gi, and by 
(106), c 2 /v 2 = — gi/gu', that is to say, v = b, as it should be. For 
then the ray falls into the wave-normal. But in general the 
ray does not coincide with the wave-normal, and so does v 
differ from D. The question whether the null-line equation 
(106) is always compatible with the electromagnetic equation 
(105) may be left to the care of the reader. If the ray be 
defined, as usual, by the Poynting flux of energy, its direction 
will be that of the vector product ITSM, and all questions 
concerning the light ray will follow from (104) with K = n as 
given by (103). 

39. Ponderomotive force, and energy tensor of the electro- 
magnetic field. The general tensors corresponding to these 
were easily suggested by the results already known from the 
special relativity theory. 

The inner product of the magneto-electric six-vector and 
the four-current, i.e., the covariant vector 

P, = F lK C\ (V) 

gives the ponderomotive force on a charge, per unit volume, 
together with its activity or, in other words, the momentum 
and the energy transferred, per unit volume and unit time, 
from the electromagnetic field upon the electric charges. 



120 Relativity and Gravitation 

In fact, using for instance cartesian coordinates and g = — 1 , 
we have for the first three components of P t , by (89) and (91), 

Pi = P T— (v 2 B 3 -v 3 B 2 ) +E^, etc., 

or if Pi, P 2 , P s be condensed into the three-vector P, 

P = P |~E + - FvB~] 

which is the familiar formula for the ponderomotive force, 
while the fourth component becomes 

P 4 = - - (E^+Em+Em) = --?- (Ev) 
c c 

or, since FvB = 0, 

Pi= - — (Pv) 

c 

which, apart from the factor — 1/c, is the activity of P. 
Somewhat more generally, the same formulae will hold with p 
replaced by p/^ — g . 

But it will be understood that from the standpoint of 
general relativity the master formula for the electromag- 
netic momentum and energy transfer is again (V), as were 
before the electromagnetic field equations, all generally 
co variant. 

By means of (IV) and (V) the four-force P L can be repre- 
sented as the covariant derivative of a second rank tensor, 
a generalization of the array of maxwellian stresses, momen- 
tum and energy density. Following Einstein's example it will 
be enough to give here the required form of P t for such co- 
ordinates for which g= — 1, and therefore, by the second 
of (IV), 

c k = ii£_- 

Thus, by (V), 

p= JL(F lK F K *)-F*^ . 



ENERGY Tensor 121 

The second term is, by the first of the equations (IV), 

p* dF^ = _ p X S dF A + dF u \ 

dx x \ dx, dx K ' 

dx L L dx l dx K J 

But the bracketed expression vanishes. In fact, since the 
summation is to be extended over all k, X, and since both /*'- 
tensors are antisymmetric, this expression can be written 

p*/dF* | *flu dF u \ 

V dx, dx K dx x / 

to be summed only over k < X. But the third term of the 
bracketed factor is -\-dF lK /dx x , so that the whole factor of 
F kX vanishes, by the first of (IV). Thus 

r^xd/^. _ , r,«x ^a _ a Ka x/s E- ^a 

t ~ » * 2 g g SaP —— , 

dx x d.r t d.Y t 

and since here k, X can be replaced by a, /3 and vice versa, 

p^ep. = -± g "o^JL(F af} F KX ) 
dx x dx t 

- "I r^ OF* **) + i *".* ^x f- GT S x ')- 
dx t dx t 

The last term can be transformed into — h F Kr F kX g xp dg />r /dx i , 
so that 

P, = -L (fl. F« x ) - | -A (F KX F* ) - 1 F* P* ^ ^r . 

Finally, if we denote by F the invariant F kX F kX and in- 
troduce the mixed tensor 

Ti =$F8] - F Ka F*\ (107) 

the last formula becomes 

p t =— " -hg KX — n , (ios) 

dx a dx l 



122 Relativity and Gravitation 

exhibiting the four-force in terms of Tf, the energy-tensor of 
the electromagnetic field. 

To recognize in the latter an old friend consider a galilean domain and 
use cartesian coordinates. Then, the g lK being constant, (108) reduces to 
the familiar equation 

dT a 

dx a 

and since, by (89), in the present case, F«z = F 23 = M\, etc., Fu= — i^ 4 = 
Ei, etc., we have 

F=F af }F a P = 2(M*-E?), 
and (107) gives 

7V = (£i 2 - W) + {Mt - W 2 ) , 

r 1 2 = r 2 1 =£i£ 2 +-Mi-M'2, etc., 
which are Maxwell's electromagnetic stress components,* further 

Ti* = 7V= - (E 2 M 3 -E3M0), etc., 

which are the negative components of the energy flux divided by c, or the 
components of electromagnetic momentum per unit volume, and finally 

which is the negatived density of electromagnetic energy. 

The right hand member of (108) can be shown to be the 
divergence of the mixed tensor T* or its contracted covariant 
derivative T" a as defined by (47a) . In fact, since for constant 

g, by (67), < V = 0, the said divergence reduces to 

T '- = ~ J {'/}**' < 42 ^ 

and since in our case Tp is symmetrical, this can be shown 
to be identical with 

Tt a =-± +*?-r Al (42c) 

dx a dx t 

where T kX = g KV Ty. On the other hand, since g* x g KV is a 
constant, to wit 5„ , we have 

f 9 J- =-gJ-£- 

dx t dx t 

[a relation to be used also in passing from (42o) to (42c)], and 
^Tensions proper being counted positive. 



Energy 1 ensor [23 

the second term of (10S; becomes identical with th< 
term of (42c). Thus 

P t = T^ = Div (77) (10 

exhibiting the four-force as the divergence of the mixed energy - 
tensor of the electromagnetic field. 

If the electric charges are under the exclusive control of the 
electromagnetic field, the total four-force P, vanishes, and we 
have 

7? ^Div (77) =0. (109) 

These four equations are perfectly analogous to the 
'equations of matter', (65), given in Chapetr IV, the 'tensor 
of matter' being now replaced by the energy-tensor of the 
electromagnetic field defined in (107). These equations 
express in either case the principles of energy and of momen- 
tum. 

Instead of the mixed tensor (107) we can introduce the 
covariant electromagnetic tensor g lv T* = T lK . If the form 
(III) of the gravitational field-equations be used, then in the 
presence of an electromagnetic field the components of the 
latter tensor (multiplied by the gravitation constant) have 
to be included in the corresponding components of the tensor 
of matter appearing in the right-hand member of those 
equations. Thus both kinds of stresses, energy', etc., con- 
tribute to the curvature tensor G lK and through k codetermine 
the gravitational field. The contributions of the electro- 
magnetic tensor components are, of course, for all technically 
obtainable fields, exceedingly small as compared with those 
due to matter in the narrower sense of the word. Theoreti- 
cally, however, the roles of the two kinds of energy-tensors 
are equivalent. 



—9 



APPENDIX. 
A. Manifolds of Constant Curvature. 



As was mentioned in Chapter III, an w-dimensional 
manifold of constant isotropic riemannian curvature K, posi- 
tive, nil or negative, is characterized by the differential 
equations (54), which can be deduced from the general 
formula (53) for the riemannian curvature.* If we put 

where R may be any constant, imaginary or real, finite or 
infinite, the said equations are 

(tX, hk) = — (g tll g Xlc - g M g XM ) , (1 10) 

to be satisfied for all t, X, k, fx. In order to pass from Riemann's 
covariant symbols to the mixed curvature tensor use (50a). 
Thus, multiplying both sides of (110) by g Xa and taking 
account of (32) , 

a. 

Einstein's tensor G^ is the contracted curvature tensor 

G« = - (8 a K g ia -Kg*). 
JKr 

The first term in the brackets is simply g x , while the second, 



*Cf. also W. Killing, Die Nicht-Euklidischen Raumformen, Leipzig 1885, 
Section 123. 

124 



Co.VSTAVl CURVATT/WB 121 

in which <5* or 1 is to be taken n times, is equal nf^ . Thus, 
for a manifold of n dimensions, of the said kind, 

^ = - n ^- 1 .^, (Ill) 

for all values of i, k. In fine, the contracted curvature tensor 
is proportional to the metrical tensor g u . For a three-space 
the constant factor is — 2/R 2 , and for a four-manifold —3 R 1 , 
and so on. Notice that we are dealing here with isotropic 
manifolds, — a remark which will be of importance in the 
sequel. 

The curvature invariant is G = g" c G u , and since g 4 * g u =;z, 
we have, by (111), 

G= _n(n-1)_ 

This justifies, in general, the name of 'mean curvature' 
mentioned in Chapter IV and given to G by some authors. 
For a three-space we have 

G=-- , (112 s ) 

R- 

and for a four-fold, provided always it were isotropic, we 
should have 

12 

G=--. (112<) 

R 2 

It was known for a long time that the line-element of a 
three-space of constant curvature l/R 2 is, in polar coordinates 

Xi, x 2 , x 3 = r, 0. 6, 

d<r*=dr*+R* sin 2 — . [j0 2 +sin 2 <t>dd-]. (113) 

R 

In fact, availing himself of (75), the reader will find for (113) , 
as the only surviving components, 

2 r 

Gn= — — , 6 , 22 = — 2 sin 2 — , G 33 = — sin 2 <t>G 2i , 
R 2 R 

that is to say> 

<?«=—!**' (uis) 

R- 



126 Relativity and Gravitation 

thus verifying (111) for the case w = 3, whence also G = 

— Q/R 2 , as above. 

Manifestly, if we took for da 2 the negative of (113), or 

2 
inverted the signs of all g ik , we should have Gu= + — ■ g,-,-. 

Now, it will be well to notice that the same is the case if we 
subtract the (113) -value of da 2 from the squared differential 
of a fourth coordinate multiplied by a constant; that is to 
say, for a four-dimensional manifold defined by 

ds 2 = dx i 2 -dr 2 -R 2 sm 2 — . [dc^'-fsin 2 ^ 2 ] (114) 

R 

we have [not (111) with w = 4 but] 

2 
G«= — (*=i,2.3); G 44 = 0, 

as the reader can verify explicitly, and therefore, 

G = ±G U =+® . 

In fine, for a four '-fold, say space-time, of the type (114) the 
three curvature components and the invariant G have the 
same values as for an isotropic three-space with changed 
signs. Notice that this result does by no means clash with 
the general equations (111) and (112). For the space-time 
determined by the line-element (114) is not isotropic with 
respect to its riemannian curvature, even if x 4 be replaced by 

V-l Xi. 

The latter line-element plays an important r61e in Ein- 
stein's recently modified theory of which a brief account will 
be given in Appendix, B. 

Consider the four-fold defined by the somewhat more 
general line-element 

ds 2 = g A dx A 2 -dr 2 -R 2 sin 2 — [d0 2 +sin 2 <^0 2 ], 

R 

where g 4 , written for g 44 , is a function of r alone. Then, with 
/z 4 = log g 4 , the only surviving G-components will be 



Gn = l (/*7 + 2/i/') - 



CONSTAM ( I K\ \ II I' I 

2 



l_'7 



!<■■ 



Goo — 



I 



sin 2 <£ 



C?88 — 



sin - — 2 Mil- 



2r 
R 



(115 



_1 G 44 = -i (&V+ 2/? 4 ") - — - n>t - 
gi R R 

whence the curvature invariant G = —G\i-\-2G 2 2 go + Gu/g*, 
with g 2 = -R 2 sm 2 (r/R), 

G= 1 _i(/ ? v+2// 4 ")-— 4 cot - . 
R* R R 

Let us now require that G should he constant (which is, at any 
rate a necessary condition for G lK : g u = const.). Then the last 
formula will be a differential equation for // 4 = log g+. Now. 
this equation can he satisfied by 

g 4 = cos 2 ar, 
where a is a constant. In fact, this assumption gives 

h'i = —2a tan ar, h" A = — 2a 2 /cos 2 ar 
and reduces the last equation to 

n ., , 4a r „ G 

2a~-\- — cot — . tanar = G — — =const., 
R R R 2 

and this equation can only be satisfied either by a = 0, i.e.. 
g4 = l, and 

R 2 

which leads to the line-element just considered, or by a — 1 R. 
i.e., gi = cos 2 (r/R), and (7 = 12 R 2 . which gives the line- 
element 

^ 5 2 = cos 2— . dx^-dr-R 2 sin 2 — [dtf+sin-Qdd*], (116) 

R R 



utilized by de Sitter. (Cf. Appendix, C, infra). The con- 



128 Relativity and Gravitation 

stant value of the invariant G is in this case 

R 2 
that is to say, apart from the changed sign, such as would 
correspond, by (112), to a genuine isotropic four-fold con- 
sidered at the beginning. Moreover, introducing g 4 = cos 2 (r/R) 
into (115), we have at once 

3 1 r 3 

G n = - — , G 22 = —— Gss= ~ 3 sin2 v > Gii = vz g4 ' 
R 2 sm z (f) R R A 

and since g\ = — 1 , g 2 = — i? 2 sin 2 (r/R) , and all components with 
i9^k vanish, 

^ = | 2 ^, (H6 1 ) 

which, apart from the changed sign of the constant factor, 
agrees with (111) for w = 4. 

On the other hand, substituting into (115) the alternative 
solution g 4 = l we have, for the line-element (114), 

2 
G u = — gu («=i,2,3) ; Gu = 0. (114 1 ) 

The best way of stating the properties of the two solutions 
is to write the corresponding contravariant tensors which in 
our case reduce to G a = GJg u . These are, for the line- 
element (114), 

QU == Q22 =G S3 = £ G 4 * = 0, (H4 2 ) 

R 2 
and for the line-element (116), 

G^ = G 22 = G ss = G ii = - . (116 2 ) 

R 2 

Thus the time-space defined by the line-element (116) 
behaves, apart from the common sign change, as an orderly 
four-fold of constant and isotropic riemannian curvature. 
This is its characteristic difference from the manifold defined 
by (114) which is deprived of isotropy and is a rather loose, 
uneven melange of time and space. Such at least would be 



Einstein's \j.\\ Equations L29 

the comparison of Einstein's line-element 1 1 1 1 1 with deSitfc 
(116), from the standpoint of general geometry. Their 

physical merit must, of course, be judged by other standards. 

B. Einstein's New Field-Equations and Elliptic Space. 

About two years after the publication of the original form 
of the gravitational field-equations, (III), Chapter IY. 
Einstein found weighty reasons for slightly modifying them.* 
Without attempting an exhaustive discussion of all his reas 
for that change or amplification we shall give here a brief 
account of his new field-equations and of some of their 
consequences. 

The tensor of matter T^ being given, the metrical and 
at the same time the gravitation tensor components g u are 
not, of course, determined by the field-equations alone, as 
indeed would be the case with any other set of differential 
equations in infinite space (and time). A necessary supple- 
ment of the data consisted, exactly as in the case of Laplace- 
Poisson's equation, in prescribing the behaviour of the g„ at 
infinity. Now, as may best be seen from the example of the 
radially symmetrical field treated in Chapter V ', the g u were 
assumed to tend 'at infinity', that is, for ever growing r L. 
to their galihan values g lK , say in cartesian coordinates, 

-10 

0-1 

0-1 

1 

But such boundary or limit conditions, not being independent 
of the choice of the coordinate system, have seemed ' repugnant 
to the spirit of the relativity principle'. In fact, to remain 
generally invariant the limit tensor would have to be an array 
of sixteen zeros. Moreover, the adoption of the galilean or 
inertial tensor at infinity would be tantamount to giving 
up the requirement of the relativity of inertia. For whereas 
the inertia or mass of a particle generally depends upon the 

*A. Einstein, Kosmologische Betrachtungen zur allgenieincn Rclativitiits- 
theoric. Berlin Sitzungsberichtc, 1917, pp. 142-152. 



130 Relativity and Gravitation 

g LK and these are even at the surface of the sun but slightly 
different from g lK , the mass of the particle at infinity would 
differ but very little from what it is near the sun or other 
celestial giants. In fine, the bulk of its mass would be inde- 
pendent of other bodies, and if the particle existed alone in 
the whole universe, it would still retain practically all its 
mass. As a matter of fact we do not know whether such 
would not be the case.* But somehow, not uninfluenced by 
Mach's older ideas, Einstein inclines to the belief that every 
particle owes its whole inertia to all the remaining matter in 
the universe. Yet another reason against the said conditions 
at infinity is given which is based on considerations borrowed 
from the statistical theory of gases and which would equally 
apply to Newton's theor)^. But for this the reader must be 
referred to Einstein's original paper (I. c, §1). 

In conclusion Einstein confesses his inability to build up 
any satisfactory conditions at infinity, in space that is.f But 
here a way out naturally suggested itself. The conditions at 
infinity being hard or perhaps impossible to find, let the 
world or universe be closed in all its space extensions. If this 
be a possible assumption, no such conditions were needed. 

Thus Einstein comes to assume space to be a finite, closed 
three-fold of constant curvature, in short an elliptic space, 
either of the antipodal (spherical) or of the polar, properly 
'elliptic', kind. But, as we saw before, the curvature proper- 
ties of space-time are modified by the presence of matter, the 
invariant G, for instance, being proportional to the density 
of matter. Thus the curvature of space, as a section of the 
four-fold, can only be approximately constant and isotropic, 
and Einstein assumes therefore that space is elliptic or very 
nearly so on the whole, deviating here and there, within and 
near condensed matter, from the average value of its curvature 
1/R 2 and from isotropy, somewhat as, in two dimensions, a 

*Provided, of course, we had some massless phantoms to serve us as 
a reference system and thus to enable us to state the lonely particle's perse- 
verance in uniform motion. 

t'Fiir das raumlich Unendliche'. There is nowhere a mention of the 
behaviour at infinite past or future, no doubt, because such questions with 
regard to time are not urgent in the usual (stationary) type of problems. 



Einstein's Ne^j i" r itkm L31 

slightly corrugated or wrinkled sphere. As we know, the lino- 
element of such a three-spare i- 

da* =dr*+R* kin* — (d^+sinfyd**), 

and Einstein constructs the line-element which is to determine 
the four-world 'on the whole' by simply subtracting da'- from 
dx A 2 = c^df 1 . 

In short, far enough from condensed matter, stars, planet-. 
and so on, his line-element, in polar coordinates, is 

ds- = dx 4 ---dr' 2 -R- sin 2 — (</<£- + si n 2 0</0-). (114) 

a differential form treated in Appendix A.* 

Now, this line-element is incompatible with Einstein's 
older field-equations (III). In fact, the corresponding curva- 
ture tensor consists of the only surviving components 

G\,= -|- , G 4 4=0; G= A, (114') 



*From the four-dimensional point of view, the assumption that three- 
dimensional space is elliptic is, of course, as unsatisfactory as the older 
assumption of galilean g lK at infinity. For although the space properties as 
defined by da 2 are invariant for transformations of the .v 1( xj, .v 3 alone into 
any x\, x'o, x' 3> they cease to be so when all four coordinates are freely 
transformed. What is then invariant are the curvature properties of the 
four-fold of which the three-space is an arbitrary section. * If at least the 
four-fold (114) were isotropic, Einstein's elliptic space could be invariantly 
defined as that of its sections to which corresponds the minimum mean 
curvature, and this is the mean curvature of the four-fold itself (cf. W. 
Killing, Inc. tit., pp. 79-83). But the four-fold defined by (114) is by no 
means isotropic, as was explained in A. Figuratively, and with some 
licence, it resembles not a sphere but rather the surface of a circular 
cylinder. By (114) not only the value of the curvature of three-space 
remains unsettled but even its property of being at all a closed space. In 
fine, the assumption that three-space is elliptic should be as 'repugnant to 
the spirit of relativity' as was the older condition at infinity. But as a 
matter of fact it did not appear to Einstein in that light. 

The clearest way of stating Einstein's new assumption is to say that, 
outside of condensed matter, it is possible to choose a coordinate system in 
which the line-element ds 2 assumes the form (114V 



132 Relativity axd Gravitation 

and if these values be introduced into the field-equations 
(Ilia), which are identical with (III), the result is 

1 _ 8tt 3 _ St „ 

But 'on the whole', that is, outside of condensed matter, T u , 
T 2 o, T 33 are to vanish (though the value of T i4: and T = p need 
not be prejudiced), and since actually g n = —1, etc., the in- 
compatibility of (114) with (III) is manifest. 

Such being the case. Einstein is driven to modify his 
original equations (III) by subtracting from their left-hand 
members the terms Xg„. with a constant X. Thus his new 
field-equations are 

G IK -\g Ui =--AT tK -^g lK T) i (117) 

c- 

and since these give, obviously, 

G-4X= — T, (117a) 



c- 



they can also be written 



5- 



G x -i(G-2\) gtK = -^ T«. (1176) 



Since the supplementary term \g lk is itself covariant of rank 
two, the general covariance of the new equations is manifest. 
It remains to evaluate the constant X in terms of the 
curvature 1/R 2 . Now, if we assumed that outside of 'con- 
densed matter' there is no matter at all, i.e., T^ = for all i, k, 
we should have, by (114 1 ) and the first of (1176), \ = 1/R 2 , 
clashing with (117a), through (114 1 ) which would require 
X = 3/i? 2 . But, as Einstein expressly states, his new theory 
is to be associated with the approximate concept that all the 
matter of the universe is spread uniformly over immense spaces. 
In other words, Einstein substitutes for the granular structure 
of the universe (the grains being not only planets but stars, 
nebulae and similar giants) a macroscopically homogeneous 
distribution of matter, exactly as for many purposes a con- 



En Xi-.u Eouatio 

tinuous homogeneous medium is substituted for an assem- 
blage of molecules or atom-. The total n taiued in 

universe being M and its volume V, Einstein's homogi • 
density, prevailing on the whole. Is 

M 

Po= — • 

V 

Only here and there, within the celestial bodies, the density p 
exceeds p considerably, and is perhaps somewhat larger in 
interstellar spaces within our galaxy than half way between 
one star cluster or 'island universe' and another, a million or 
more light years apart. Moreover, basing himself upon the 
known fact of the small relative velocity of stars as compared 
with the light velocity, Einstein makes the approximate 
assumption that there is a coordinate system, relatively to 
which matter is on the whole permanently at rest, and in 
which therefore the tensor of matter is reduced to its 44-th 
component which is then also its invariant T = p. 
In fine, we have outside of condensed matter 

T ii =T = p 

as the only surviving component, and therefore, by (414 
and (1176).' 

1 4- 

Thus, Einstein's new field-equations 117' become ulti- 
mately 

G m -l(c-Q 5lK =- ??r„. (117c) 

At the same time we see that the curvature of space on the 
whole is proportional to the average density of matter. 

1 4tt 
^ = ^p . 118 

The whole volume of elliptic space of the polar or properly 
elliptic kind being 



134 Relativity and Gravitation 

the total mass of the universe, in astronomical units, will be 

M=—R, (119) 

4 

which moved some authors to the enthusiastic exclamation: 

'the more matter, the more room'. The corresponding 

'gravitation radius', or better, the mass in bary -optical units, 

which is a length, would be 

L— — = — , (119a) 

c 2 4 

or just one-quarter of the total length of an elliptic straight 
line.* 

According even to our coarse knowledge of the average 
density of matter (some thousand suns per cubic parsec), and 
in view of the formula (118), it is impossible to believe in a 
curvature radius much smaller than 10 12 astronomical units 
or, say, R = 10~° kilometers. This would mean, by (119a), a 
total mass amounting again, in bary-optical units, to almost 
10 20 kilometers. To this tremendous total our own sun contri- 
butes but 1| kilometers, and our whole galaxy not more than 
10 10 kilometers. The total would thus require 10 10 such galaxies 
or Shapley's 'island universes'. All these stellar systems may 
perhaps be found among the spirals. But if Shapley's esti- 
mate (Bull. Nat. Res. Council, 1921, No. 11, The Scale of the 
Universe) be materially correct, these island universes are 
from 500 thousand to 10 million light years apart, and then 
it remains to be seen whether the last mentioned space would 
be ample enough. Yet it would certainly be foolish to deny 
the possibility of a much larger R and of the existence of 
many more island universes. That Einstein's requirement, 
at least in the present state of astronomical knowledge, can 
at any rate be satisfied, is perhaps best seen from its form 
(118) which is compatible with as small an average density 
as we please. 

*The total length of a straight line (geodesic) in the polar kind of space 
is ttR, and in the antipodal or spherical kind of space 2ttR. The total 
volume of the latter space is 2t 2 R s , which would give the double mass, as 
in Einstein's paper. The space in question being thus far defined only 
differentially, the choice between the polar and antipodal kind remains free. 



Sitter's Space-Time 135 

Further details concerning these cosmological speculations 

will be found in de Sitter's third paper on Einstein's Theory 
of Gravitation,* where the role played by elliptic 9pace in 

astronomy since the time of Schwar/.schild (1900) is discussed. 
The light rays corresponding to Einstein's line-element 
(114) turn out to be straight lines in elliptic space, and these 
lines, described with uniform velocity, are also tin orbits of 
free particles. Planetary motion would undergo some modi- 
fications due to the finite value of R; but these are, for the 
present, too small to be detected. Nor does Einstein's 
'cosmological term', as the supplement f^/J? 2 to his original 
field-equations is called, lead to any other predictions verifi- 
able in our days by experiment or observation. 

C. Space-Time according to de Sitter. 

Returning to Einstein's amplified field-equations (117) 
let us assume, with de Sitter, that there is outside of 'con- 
densed matter' no matter at all, so that in such domains all 
the components of T tK , including T i4 , vanish. Thus we shall 
have, in free space, so to speak, 

for all l, k. Now, as we saw in Appendix B, these equation- • 
which are of the form of (111), can be satisfied by the line- 
element (116), and give G = 12/R 2 . And since, on the other 
hand, 

we have _ 3 

~ R 2 ' 

This is the solution of the cosmological problem proposed 
by de Sitter in his last quoted paper. Thus, de Sitter's free 
space-time is defined by the line-element 

ds 2 = cos-- c-df 2 — dr* - R 2 s'ui 2 --[d4>' 2 + sm 2 4>dd 2 ] (116) 

R R 

and is therefore, as we saw, a manifold of constant isotropic 

*W. de Sitter, Monthly Notkes of R.A.S., November 1917. 



136 Relativity and Gravitation 

curvature. Within matter Einstein's new equations, with 
X = 3/i? 2 , are valid, i.e., 

G lK -^-=--(T lK -^g lK T). (120) 

R z c l 

The isotropy of de Sitter's space-time, expressed by 

R 2 
as in (116 2 -), distinguishes it characteristically from Einstein's 
space-time. This goes hand in hand with p = outside of 
matter proper. 

De Sitter's line-element differs from Einstein's by 

g 4 4 = COS 2 — 

instead of gu = 1- Consequently, if the permanency of atoms 
be assumed as in Chapter V, the spectrum lines of distant 
stars should be displaced towards the red. If r be the distance 
of a star from an observer placed at the origin of coordinates, 
the observed wave-length should be increased from 1 to 

1 : cos — , becoming infinite for r = - — R, the greatest distance 
R 2 

possible in a properly elliptic space. Manifestly, everything 
is at a standstill at the equatorial belt, i.e., all along the polar 
of any observing station as pole. This, though sounding 
strangely, entails no actual difficulty at all. As to the spec- 
trum shift of less distant celestial objects, de Sitter quotes 
the helium or 5-stars which show a systematic displacement 
towards the red such as would correspond to a velocity of 
4'5 km. per sec. If, as de Sitter suggests, one-third of this 
is considered as a gravitational Einstein-effect, the remainder 
may be accounted for by the decrease of £44, and since the 
average distance of the I?-stars is believed to be r = 3.10 7 
astronomical units, we should have 

3 - 1()7 in -6 
1— cos =10 5 , 

R 
and therefore a curvature radius i? = §10 10 . But there is, 
for the present, nothing cogent in the attribution of the said 



Gravitation and El» rao 137 

remainder of spectrum shift to the dwindling oi g« witii dm 
distance, and ii would certainly be premature either to n 
or to accept the results of this attractive piece of reasoning. 

Other consequences of the theory and a more thorough 
comparison with Einstein's solution will be found in de- 
Sitter's paper. Here it will be enough to mention still that 
according to de Sitter's line-element the parallax of a star 
should reach a minimum at t = \tR, the greatest distance in 
the polar kind of space (which de Sitter prefers to the anti- 
podal). This minimum, of the semi-parallax, is equal to 
p = a/R, if a be the distance of the earth from the sun. On 
the other hand, Einstein's line-element gives, for r = ?,7i\ff, a 
vanishing parallax. Since de Sitter's minimum is at least as 
small as £ = 10 -10 = 0"a00002, one cannot reasonably hope to 
discriminate between the two proposals by direct observations 
of parallaxes, while indirect ones contain too many assump- 
tions to be considered as crucial. 

Soon after the publication of de Sitter's paper Einstein 
raised some objections to his form of the line-element. For 
these, however, not altogether crushing, the reader must be 
referred to Einstein's own paper (Berlin Siizungsberichte, 
March 1918, pp. 270-272). 

D. Gravitational Fields and Electrons. 

The problem of the equilibrium of electricity constituting 
the electron as the structural element of matter proper, 
already attacked by G. Mie and others, has been taken up 
by Einstein in a paper of April 1919 (Berlin Sitzungsber., 
pp. 349-356). The result of the investigation is that this 
tempting question cannot be completely answered by means 
of the field -equations alone. For details of the reasoning 
the reader must be referrred to the original paper. It will 
be enough to mention that the fixed relation between the 
universal constant X in the amplified field-equations and the 
total mass of the universe, as related in Appendix B, is here 
given up. Space continues to be considered as closed but the 
curvature radius R and, therefore, the volume of the universe 
appears as independent of the total mass contained in it. 
though its macroscopic density p is still treated as uniform. 



INDEX 



(The numbers refer to the pages; 



Abraham, M 

Absolute, Cayley's 51,52 

Absolute differential calculus. . 39 
Absolute value, or size, of vec- 
tor 51 

Angle 50 

Antipodal kind of elliptic space 6 

Antisymmetrical tensors 44 

Associated invariant 53 

vectors 54 

Astronomical unit of mass .... 74 

Atoms, as natural clocks 104 

Bary-optical unit of mass 134 

Bessel 10 

Bianchi, L 28, 39, 64, 71 

Boundarv conditions 129 



Canal rays 

Cayley 5 

Centrifugal acceleration 3 

Christoffel 

symbols 

Clifford 

Closed space 

Coelostat distortions 

Compatibility conditions 

Componens of a tensor 

Conj ugate vectors 

Conservation laws 

Constant curvature 

Constant light-velocity, prin- 
ciple 

Contraction, of mixed tensors. 

Contravariant devriative 

tensor, denned . 

Cosmological term 

Covariance of natural laws. . . . 

Covariant differentiation 

tensor, defined 

Current, four- 

Curvature, gaussian 1 

riemannian 



105 

1,52 

1,37 

59 

27 

13 

130 

102 

118 

41 

54 

87 

67 



46 

59 

41 

135 

23 

59 

41 

109 

',65 

67 



Curvature invariant ... 

Curvature tensor 63,68 

contracted 70 

Cyanogen spectrum lines 104 

Deflection of light rays 100-102 

Density of matter 134 

Differential geometry 5 

quadratic form. . . 14 
Differentiation of tensors . . 59 et seq. 
Divergence of mixed tensor. . . 83 

of a vector 62 

of a six-vector .... 62 

Eclipse expeditions 102 

Eddington, A. S 39, 80, 103 

Einstein, 1, 10, 11, 12, 19, 22, 24, 28, 

38, 39, 56, 58, 61, 70, 77, S2, 86, 

88, 104, 107, 113, 129, 130, 137 

Electro-magnetic six-vector . . . 109 

Electromagnetic equations .... 106 

et seq. 
stress, momentum, 

and energy 75 

Electrons 137 

Electrostatic potential 112 

Elementary flatness 13 

Elevator 11 

Elliptic space 6, 130 

Energy, principle of 86-87 

Energy tensor 75, 122 

Eotvos, R 10 

Equation of motion, of a free 

particle 28 

Equivalence hypothesis 12 

Equivalent differential forms. . 16 

Expansion, of a vector 59 

of a six- vector .... (52 

Fermat's principle 100-101 

Field equations, gravitational 

70. 77. $9. 132 



140 



Index 



Fixed-star system 1 

Four-current 109 

Four-index symbols 17, 64 

Four-potential Ill 

Four- vector 4, 40 

Fundamental quadratic form . . 52 

tensor 52 

Free particle motion, and geo- 
desies 8, 20 

Galaxies 134 

Galilean coefficients 6, 129 

Galileo 10 

Gaussian coordinates 39 

General relativity principle ... 22 

Geodesies 7, 26-28 

Gradient, of a scalar field 49 

Gravitation, and Christoffel 

symbols 29 

Gravitation law, Newton's. ... 73 

Gravitation radius 95 

Gravitational field equations, 

69 et seq., 89, 132 
waves 90 

Hamiltonian principle 88 

Heavy and inert mass 10 

Helium or B-stars 136 

Hilbert, D 88 

Holonomous transformations. . 16 

Homaloidal, or flat, manifold . . 67 

Hydrogen nucleus 80 

Indices, upper and lower 45 

Inertia, induced 130 

of energy 74 

Inertial systems 1 

Inner product, of tensors 42 

Invariance, of line-element. ... 16 

Invariants 42, 46, 47 

metrical 53,57, 

62, 79, 88, 125 

Island universes 134 

Isotropic curvature 67, 125 

Jacobian 15 

Jeffreys, H 99 

Jewell, E. L 104 



Keplerian laws 96, 98 

Killing, W 64, 124, 131 

Kottler, F 107 

Laplace- Poisson's equation. .69, 73, 

74,79 

Laue, M. v 75,76 

Law of (algebraic) inertia 16 

Levi-Civita, T 39, 41,101 

Light propagation, and mini- 
mal lines 8,20 

Line-element 5 

Linear differential form 113 

Lipschitz's theorem 66 

Local coordinates 12 

Lor, matrix 75 

Lorentz, H. A 88 

Lorentz transformation 4 

Mach, E 130 

Magneto-electric six- vector . . . 107 
Mass, astronomical unit of. ... 74 

Matter : 74, 75, 77 

equations 82,85,123 

Maxwell's electromagnetic 

stress 122 

equations 106 

Mean curvature 79, 125 

Mercury's perihelion motion. . 99 

Metrical manifold 50 

properties of tensors . 53 

Mie, G 10,137 

Minimal lines 7 

Minkowski 4,75 

Mixed tensor, defined 45 

Momentum, principle of 86-87 

Mosengeil, K. v 74 

Natural clocks 104 

volume 58 

Newcomb G9 

Newton 10 

Newton's equations of motion. ."6 

Node motion, of Venus 99 

Non-holonomous transforma- 
tions 14 

Norm, of a vector 53 



Index 



MI 



Outer product, of tensors. 13 

Orthogonal coordinates 113 

vectors, define I. 56 

Parallax 137 

Perihelion motion 95 et seq. 

Permanent field 70 

Perturbations, secular 99 

Planetary motion 95 6/ seq. 

Polar kind of elliptic space ... 6 
Ponderomotive force, in elec- 
tromagnetic field 1 H> 

Potential, electrostatic 112 

four- Ill 

newtoniau 36 

retarded 90 

vector- 112 

Poynting 74, 75 

Principe, eclipse expedition . . . 102 
Principles of momentum and 

energy .- 86-87 

Product, inner 42,48 

outer 43 

Propagation of gravitation . ... 90 
Proper time 103 

Radially symmetrical field . 92 et seq. 

Radius, gravitation- 95, 134 

of world-curvature . . . 80-81 

Rank ,of a tensor 40 et seq. 

Rankine, A.O., and Silberstein, 117 

Reduced tensor 50 

Retarded potential 90 

Reference frameworks . A, et passim 
Relativity principle, general. . . 22 

special 2 

Ricci, G 39,41 

Riemann 17, 51 , 64 

Riemannian manifold 50 

Riemann-Christoffel tensor. .62,63 

Rotating system 30-35, 101 

Rotation, of a covariant vector 61 

Russell, H. N 102 

Rutherford 81 

Scalar, tensor of rank zero .... 42 

Scalar product, of tensors 42 

Schwarzschild, K 95, 135 



,-r 

Shapley . . 134 

Shift of spectrum lines. L02-106, 138 

Sitter, W.de .33,99, 101, 127, 13ft, 

138, 137 

Six- vector 44 

Size, of a vector 51 

Sobral, eclipse expedition \0l 

Space-like vector 

Special relativity, recalled ... 1-9 
Spectrum shift, due to gravita- 
tion 102-10.'. 

due to distance 136 

Spherical space 6 

St. John, C. E 104 

Stress-energy tensor 75 

Sum of tensors 41 

Sun, gravitation radius of.. .95, 134 

Supplement of a tensor 55 

Sylvester 16 

Symmetrical tensors 4". 

Tangential world 13 

Tensors 39 et seq. 

Tensor character, criterion of.. 48 

Tensor of matter 76, 78 

Thirring, H 32 

Time-like vector 53 

Universe, mass and volume of . 134 

Vector 40 

Vector potential 112 

Velocity of light 25, US 

Venus, motion of nodes 99 

Volume 58 

Wave, electromagnetic, in gra- 
vitation field 118 

Waves, gravitational 90 

Wave surface 26 

Water, curvature in 80 

Weight and mass 10 

Weyl, H 39, SS, 103,113 

World curvature 80 

vector 4 

Wright, J. E 39 



Zodiacal matter 



99 



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